Q1. Jerry has wealth of $60 and derives utility from this according to the utility function U (w) = 1 - Where w is his wealth. Jerry now finds a lottery ticket (the drawing takes place the next day) that offers a 50% chance of winning $5. a) What is the expected value of Jerry if he takes the lottery ticket? (pay attention, it's not jerry's wealth) b) What is the minimum amount for which Jerry would be willing-to-sell the ticket? (Hint: sets a price of p, and at the minimum amount, the expected utility of selling and not selling should be the same) c) Which is bigger, your answer to (a) or (b), and suggest whether Jerry is a risk averse person based on the previous conclusion? d) If he does not sell the ticket, what is Jerry's cost of risk? (The cost of risk is the difference between the expected wealth and the certainty equivalence)

Q1. Jerry has wealth of $60 and derives utility from this according to the utility function U (w) = 1 - Where w is his wealth. Jerry now finds a lottery ticket (the drawing takes place the next day) that offers a 50% chance of winning $5. a) What is the expected value of Jerry if he takes the lottery ticket? (pay attention, it's not jerry's wealth) b) What is the minimum amount for which Jerry would be willing-to-sell the ticket? (Hint: sets a price of p, and at the minimum amount, the expected utility of selling and not selling should be the same) c) Which is bigger, your answer to (a) or (b), and suggest whether Jerry is a risk averse person based on the previous conclusion? d) If he does not sell the ticket, what is Jerry's cost of risk? (The cost of risk is the difference between the expected wealth and the certainty equivalence)

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.8P

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2. Alice believes that her car would cost £12500 to replace if it was stolen or damaged. Based on crime statistics for the area she lives in, she believes that the probability of her car being stolen or damaged is 0.15. (i) Alice's utility function is given by U(w) = ln(w) for w > 0 and she as £35000 in the bank. Calculate how much Alice would be prepared to pay (in a single payment) to insure her car against theft or damage (ii) Repeat the calculation in the previous part but now assume Alice has £500000 in the bank.
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5. Consider a decision-maker who expects to have a car accident with chance ; if this occurs, he will incur $L in damages. He can purchase as much auto insurance, q, as he likes, at a price of p per dollar of coverage: this means that if he pays pq upfront (as the insurance premium), he'll receive a payment of q from the insurance company if an accident occurs. (a) Write out his expected utility from purchasing insurance level q, assuming a utility-of- wealth function u(w) and initial wealth wo. 1 (b) Show that the optimal level of insurance, q, solves u' (wo - L + (1 - p)q)) u' (wo - pq) = p(1 - π) T(1-P) (c) Now assume that u(w) 1 - e-aw, where a > 0 (this is known as a "CARA", or "constant absolute risk aversion", utility function; a parametrizes risk aversion). Solve explicitly for the optimal insurance, and show that it does not depend on wealth.
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4. Kate has von Neumann-Morgenstern utility function U(x1,x2) = m7. She currently has $2025. a. Would she be willing to undertake a gamble that involves a gain $2875 with probability + and a loss of $1125 with probability ? Show your work and explain your answer. b. Would she be willing to undertake a gamble that involves a gain $2599 with probability and a loss of $800 with probability ? Show your work and explain your answer.
4. Marvin is given a number of choices between various lotteries where the possible outcomes include a vacation, a motorbike, a pair of shoes and a round of applause, or X = {v,m, s, a}. It is known that, according to Marvin's preferences, v > m > s > a. In addition, he claims he would be indifferent between the motorbike and a lottery that results in a vacation with probability 0.4 and a pair of shoes with probability 0.6, and he would furthermore be indifferent between a pair of shoes and a lottery that results in a motorbike with probability 0.5 and a round of applause with probability 0.5. Marvin has a utility function u: X → R (in other words u maps each element in X to a real number) and he in an expected utility maximiser (in other words his preferences conform to the axioms G1 to G6). 4.1. Assuming Marvin's claims about indifference are correct, derive u(m) and u(s) in terms of u(v) and u(a). 4.2. Von Neumann-Morgenstern utility functions are only unique up to positive affine…
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4. Show how to construct the reference dependent utility function for two friends Kate and Mary whose gains and losses are listed as follows : Kate's net worth is $ 4.5 million ( decreased from $ 5.5 to $ 4.5 million ) Mary's net worth $ 3.2 million ( increased from $ 3 to $ 3.2 million ) ( First determine the reference point ( use a parameter ) and then derive reference utility function for each ) .
At a different table, Juan wins $600 in a blackjack game. Similarly, he has to choose between $600 or the chance to play a new game. In this game, Juan has a 60% chance of winning nothing and a 40% of winning $1,000. The following graph presents the utility function of Juan with respect to money: U(w) U(w) U(1,000) U(700) U(600) U(400) 400 600 700 1,000 w 10.2.2 1.0 point possible (graded, results hidden) By how much money would his winnings need to increase or decrease so that Juan is indifferent between the $600 and the new game? (in case of an increase, insert a positive number; in case of a decrease, insert a negative number).
Your utility function is given by M1/2. You have $100 and are planning to invest in a venture where you can win or lose 50 with equal probability. Will you accept the venture? What is the minimum gain you need to make in the good scenario such that you will invest in the venture?