3. Suppose Chelsea has the following utility function over wealth: v (y) = ln y %3D (a) Show that Chelsea is risk averse. (b) Consider the lottery that yields the outcome $2 with probability ; and the outcome $8 with probability . How much money would you need to offer Chelsea to buy the lottery from her if the lottery is all she owns? (c) What is the Chelsea's risk premium for this lottery? (d) Suppose Antonio has utility function over wealth given by Va (y) = Vy and suppose Dillon has the following utility function over wealth:

3. Suppose Chelsea has the following utility function over wealth: v (y) = ln y %3D (a) Show that Chelsea is risk averse. (b) Consider the lottery that yields the outcome $2 with probability ; and the outcome $8 with probability . How much money would you need to offer Chelsea to buy the lottery from her if the lottery is all she owns? (c) What is the Chelsea's risk premium for this lottery? (d) Suppose Antonio has utility function over wealth given by Va (y) = Vy and suppose Dillon has the following utility function over wealth:

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.7P

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5 Expected Utility formulation was initially proposed as a solution to the St. Petersburg paradox (or, its predecessor). However, does it really solve all such paradoxes? More specific, consider an individual whose “little Bernoulli” utility functions is, a la Cremer, given by u(x) = x1/2. Construct a lottery similar to St. Petersburg lottery in that your lottlery, too, gives a finite prize with probability one, but not only the expected value, but also the expected utility of your lottery (calculated using the u(.) above) is not finite. Discuss how that violates EU as a solution for the St. Petersburg paradox.
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)
4. The preferences of agents A and B are representable by expected utility functions such that uA(x) = 5x^1/3 +30, and uB(x)= 1/5x - 20. Then, the following allocation of the expected returns of a risky joint investment of A and B as represented by lottery L = ((2/3);1500), (1/3);120)) is Pareto efficient: (a) xA = (500,100), xB = (1000,20) (b) xA = (100,100), xB= (1300,20) (c) xA= (80,80), xB = (1420,40) (d) xA = (750,60), xB= (750,60) (e) NOPAC
6) For the payoff table below, the decision maker will use P(s1) = .15, P(s2) = .5, and P(s3) = .35. s1 s2 s3 d1 -5000 1000 10,000 d2 -15,000 -2000 40,000 (a) What alternative would be chosen according to expected value? (b) For a lottery having a payoff of 40,000 with probability p and -15,000 with probability (1-p), the decision maker expressed the following indifference probabilities. Payoff Probability 10,000 .85 1000 .60 -2000 .53 -5000 .50 Let U(40,000) = 10 and U(-15,000) = 0 and find the utility value for each payoff. (c) What alternative would be chosen according to expected utility?
3. Suppose that individuals with a strong enthusiasm for technology are willing to pay $500 now for the latest iPhone, but only $350 if they must wait a year. Normal people are willing to pay $250, and their desire to purchase does not vary with time. Assume for simplicity that there are equal numbers of each customer type (strong enthusiasm and normal), that the MC of the iPhone is $100, ignore the time value of money, and there are only two years. (Hint: With discrete problems consider all the different options and evaluate each of them to determine the optimal strategy. In this case, there are only three prices given and two time periods, so a total of 6 pricing strategies exist but some of them will.) What is the highest profit level that can be achieved if the price for an iPhone is fixed for both years (uniform pricing)? What is the highest profit level that can be achieved if the price for an iPhone can be altered across the two years?
Givenu(x)=x0.5 Lottery A Probability 0.50 0.25 0.25 Outcome 64 16 0 For automatic grading, give all numerical answers to exactly two decimal places. Do not include currency signs 1) What is the expected value? (Give the answer as 36.00, not 36) 2) What is the expected utility? 3) What is the certainty equivalent? (Number only) 4) What is the risk premium? 5) Would this person rather receive 20 for sure than play Lottery A? (Answer should be Y or N for auto-grading to work) 6) (Harder) In many applications of expected utility, it is possible to lose money. The usual way of handling this is to interpret utility in terms of final wealth. Suppose it costs money to play this lottery. If starting wealth is 100, calculate the expected utility of playing lottery A if the price of playing is 15. Your answer should be to two decimal places. (Note: calculating the certainty equivalent of the lottery would be a little different than we've done in class. Squaring your EU result would give…
Prospect Y = ($6, 0.25 ; $15, 0.75) If Will's utility of wealth function is given by u(x)=x0.25, what is the value of CE(Y) for Will? (In other words, what is Will's certainty equivalent for prospect Y?) (The certainty equivalent represents the maximum amount a person would be willing to pay to acquire a risky prospect, and equivalently, the lowest price for which they would be willing to sell a risky prospect if they already owned it) (Note: The answer may not be a whole number; please round to the nearest hundredth) (Note: The numbers may change between questions, so read carefully)
You are evaluating the possibility that your company bids $150,000 for a particular construction job. (a) If a bid of $150,000 corresponds to a relative bid of 1.20, what is the dollar profit that your company would make from winning the job with this bid? Show your work. (b) Calculate an estimate of the expected profit of the bid of $150,000 for this job. Assume that, historically, 55 percent of the bids of an average bidder for this type of job would exceed the bid ratio of 1.20. Assume also that you are bidding against three other construction companies. Show your work.
5) Factoring in the market price of oil (as a random event, depicted by a circle) to determine revenues. Planners might use a continuous random variable, like a normal distribution with a known mean and standard deviation. They also might use a limited set of discrete values to approximate a continuous random variable. For this example, we'll assume that, for the expected distribution of oil prices, the planners would select an extraction option giving an expected payoff of $120M. Note:- Do not provide handwritten solution. Maintain accuracy and quality in your answer. Take care of plagiarism. Answer completely. You will get up vote for sure.
a. Suppose that you took part in a lottery that has a chance to increase, decrease or have no effect on your level of income. With probability 0.5, your income remains at it original level K500; with 0.2 probability, your income increases to K700; and with probability 0.3, your income decreases to K400. The utility function is.u（1） =I^0.7where I denote income leveli.Using the utility function show that the consumer's risk preference is averse. (2marks)ii.Calculate both the EU and EV of the income. (4marks)iii.Using the results in (il) above, indicate the attitude to risk of this consumer. (2marks)
Suppose Caroline is choosing how to allocate her portfolio between two asset classes: risk-free government bonds and a risky group of diversified stocks. The following table shows the risk and return associated with different combinations of stocks and bonds.CombinationFraction of Portfolio in Diversified StocksAverage Annual ReturnStandard Deviation of Portfolio Return (Risk)(Percent)(Percent)(Percent)A 0 1.50 0B 25 3.00 5C 50 4.50 10D 75 6.00 15E 100 7.50 20There is a relationship between the risk of Caroline's portfolio and its average annual return.Suppose Caroline currently allocates 75% of her portfolio to a diversified group of stocks and 25% of her portfolio to risk-free bonds; that is, she chooses combination D. She wants to reduce the level of risk associated with her portfolio from a standard deviation of 15 to a standard deviation of 5. In order to do so, she must do which of the following? Check all that apply. Sell some of her stocks and use the proceeds to purchase…
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…