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 lotteryWhat 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 equivalence for this lottery? Suppose you have to pay $2 for a ticket to enter a competition. The prize is $19 and the probability that you win is . You have an expected utility function with u(x) = log x and your current wealth is $10.What is the CE of this competition?What is the risk premium?Should you enter the competition? Adam has just purchased a new car and has to decide whether to buy insurance to cover his new car in the event of a loss. Assume that Adam knows the probability (p) of him having an accident and loosing his new car. The car is valued at L and the amount of insurance to purchase for this value is X. Adam’s entire wealth after buying the car is W. Let r be unit price of insurance.Briefly explain the problem of the insurance company and show that for insurance to be actuarially fair, the premium must equal the probability of accident. Suppose Adam’s preferences is represented by a VNM utility function and insurance is actuarially fair, how much insurance will he buy? Now suppose Adam can influence p by installing an alarm on his car to reduce the probability of loss. The cost of installing the alarm is given as , where .Show that the optimal probability of loss will be . Write out the new expected utility of Adam and show that as a risk neutral individual his probability of accident will be unity if he buys full insurance. Clearly show that given your results in (ii), the individual will have no incentive to buy insurance (Hint: use the expected utility of the individual).

Since you have posted multiple questions with multiple sub parts, we will provide the solution only…

Answered: Suppose you have to pay $2 for a ticket…

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.8P

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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 equivalence for this lottery?

Suppose you have to pay $2 for a ticket to enter a competition. The prize is $19 and the probability that you win is . You have an expected utility function with u(x) = log x and your current wealth is $10.

What is the CE of this competition?
What is the risk premium?
Should you enter the competition?

Adam has just purchased a new car and has to decide whether to buy insurance to cover his new car in the event of a loss. Assume that Adam knows the probability (p) of him having an accident and loosing his new car. The car is valued at L and the amount of insurance to purchase for this value is X. Adam’s entire wealth after buying the car is W. Let r be unit price of insurance.

Briefly explain the problem of the insurance company and show that for insurance to be actuarially fair, the premium must equal the probability of accident.
Suppose Adam’s preferences is represented by a VNM utility function and insurance is actuarially fair, how much insurance will he buy?
Now suppose Adam can influence p by installing an alarm on his car to reduce the probability of loss. The cost of installing the alarm is given as , where .

Show that the optimal probability of loss will be .
Write out the new expected utility of Adam and show that as a risk neutral individual his probability of accident will be unity if he buys full insurance.
Clearly show that given your results in (ii), the individual will have no incentive to buy insurance (Hint: use the expected utility of the individual).

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