Let W represents an individual's annual earned income and U(W) = (W/10)0.5 is this individual's von Neumann- Morgenstern utility index (or utility function). This individual earned income is $49,000. This individual faces the prospect of a 20% chance of needing health care, with a price tag of $13,000. Assume this person is risk averse. Also assume that the insurance company has only claim costs and that administrative costs are $0. The maximum health

Let W represents an individual's annual earned income and U(W) = (W/10)0.5 is this individual's von Neumann- Morgenstern utility index (or utility function). This individual earned income is $49,000. This individual faces the prospect of a 20% chance of needing health care, with a price tag of $13,000. Assume this person is risk averse. Also assume that the insurance company has only claim costs and that administrative costs are $0. The maximum health

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter18: Asymmetric Information

Section: Chapter Questions

Problem 18.5P

See similar textbooks

Similar questions

Dr. Gambles has a utility function given as U(w)=In(w). Due to the pandemic affecting his consulting business, Dr Gambles faces the prospect of having his wealth reduced to £2 or £75,000 or £100,000 with probabilities of 0.15, 0.25, and 0.60, respectively. Suppose insurance is available that will protect his wealth from this risk. How much would he be willing to pay for such insurance?
Suppose an individual earns income $600 when they are sick, and $1000 when they are healthy. Suppose this individual is sick with probability p = 0.5 and has a utility function over income, I, of U(I) = ln(I). Is this individual risk-averse, risk neutral or risk-loving? Suppose she is able to purchase insurance at any amount from at an actuarially fair price. Fully describe the amount she would purchase (payout, premium and final outcomes). Verify that she is better off with the contract in part b, as opposed to being uninsured. Suppose insurance company A offers a payout q = $400 (when she is sick) at a premium of r = $220 and insurance company B offers a payout of $200 at a premium of $100. Company A's contract is: Actuarially fair or unfair? Is it full or partial insurance? Company B's contract is: Actuarially fair or unfair? Is it full or partial insurance? Which contract does this individual prefer? Suppose contract A is unfair, but offers full coverage at price . Contract B…
Abigail is a consumer whose utility is a function of her total wealth W. u(W ) = log W. Suppose that Abigail begins with initial wealth of A = 100 but will suffer a serious illness with probability π = 0.15 which will require extensive treatment costing L = 80. To hedge against this risk, Abigail considers buying a health insurance policy. She may buy as much insurance I as she wishes at a cost of p per dollar of coverage, so her payoffs in each state are Healthy Ill Probability 0.85 0.15 No Insurance 100 20 Claim 0 I Premium −pI −pI a) Show that Abigail is risk averse. b) Suppose that the insurance premiums are actuarially fair so that p = 0. Find Abigail’s expected wealth E[W ] and expected utility E[u(W )] as functions of how much insurance she buys I. c) How much insurance should Abigail buy?
Abigail is a consumer whose utility is a function of her total wealth W. u(W ) = log W. Suppose that Abigail begins with initial wealth of A = 100 but will suffer a serious illness with probability π = 0.15 which will require extensive treatment costing L = 80. To hedge against this risk, Abigail considers buying a health insurance policy. She may buy as much insurance I as she wishes at a cost of p per dollar of coverage, so her payoffs in each state are Healthy Ill Probability 0.85 0.15 No Insurance 100 20 Claim 0 I Premium −pI −pI a) Now suppose that the insurance company raises premiums to p = 0.2 so that they are no longer actuarially fair. Find Abigail’s expected wealth E[W ] and expected utility E[u(W )]. b) How much insurance should Abigail buy now?
Leora has a monthly income of $20,736. Unfortunately, there is a chance that she will have an accident that will result in costs of $10,736. Thus leaving her an income of only $10,000. The probability of an accident is 0.5. Finally assume that her preferences over income can be represented by the utility function u(x) = 2ln(x).a) What is the expected income? What is Leora’s expected utility (you may leave in log form)? b) What is the certainty equivalent to her situation? What is the risk premium associated with her situation?c) What is the maximum that Leora would be willing to pay for a full insurance policy?d) Illustrate her expected utility, expected wealth, certainty equivalent, the risk premium and her willingness to pay for a full insurance policy in a diagram.
Tess and Lex earn $40,000 per year and all earnings are spent on consumption (c). Tess and Lex both have the utility function c. Both could experience an adverse event that results in earnings of $0 per year. Tess has a 1% chance of experiencing an adverse event and Lex has a 12% chance of experiencing an adverse event. Tess and Lex are both aware of their risk of an adverse event. 1. Suppose the actuarially fair premium charge is 2600, Calculate Tess’ expected utility with full insurance if she is charged the premium. Round to two decimal places. 2. What is the premium that private insurance companies will charge for full insurance? Round to two decimal places. 3.Assume the social welfare function is the sum of the Tess’ and Lex’s utility functions. Select the correct statement regarding the explanation for what has happened in the private market and the role of social insurance. a.Adverse section has lead to market failure. The government could improve social welfare by…
Priyanka has an income of £90,000 and is a von Neumann-Morgenstern expected utility maximiser with von Neumann-Morgenstern utility index u(x) = square root x . 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
A risk-averse expected-utility maximizer has initial wealth w0 and utility function u. She facesa risk of a financial loss of L dollars, which occurs with probability π. An insurance companyoffers to sell a policy that costs p dollars per dollar of coverage (per dollar paid back in theevent of a loss). Denote by x the number of dollars of coverage.(a) Give the formula for her expected utility V (x) as a function of x.(b) Suppose that u(z) = −e−zλ, π = 1/4, L = 100 and p = 1/3. Write V (x)using these values. There should be three variables, x, λ and w. Find the optimal value of x,as a function of λ and w, by solving the first-order condition (set the derivative of the expectedutility with respect to x equal to zero). (The second-order condition for this problem holds butyou do not need to check it.) Does the optimal amount of coverage increase or decrease in λ,where λ > 0?(c) Repeat exercise (b), but with p = 1/6.(d) You should find that for either (b) or (c), the optimal coverage…
Find the Pratt - Arrow risk - aversion function for a utility function U(W) = log(0.5-W + 500), where W is the amount of wealth in €. Suppose that an investor's wealth is subject to outcomes -800 €, 500 €, 500 € and 1, 000 € which affect the initial amount of 2,500 € with probabilities of their occurrence 40%, 15%, 15% and 30%, respectively. a) Using the Taylor approximation to certainty equivalent, calculate an approximate expected utility value. b) Calculate the certain equivalent of the investor's uncertain wealth. Interpret.
. Priyanka has an income of £90,000 and is a von Neumann-Morgenstern expected utility maximiser with von Neumann-Morgenstern utility index u(x) = square root x. 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?
Suppose that there is a 20% chance Malik is injured and earns $100,000, and an 80% chance he stays healthy and will earn $500,000. Suppose further that his utility function is the following (utility = square root of income) Malik is risk ____. He will prefer ____ (given the same expected income). a. lover; actuarially fair and full insurance to no insurance b. averse; no insurance to actuarially fair and full insurance c. neutral; he will be indifferent between actuarially fair and full insurance to no insurance d. lover; no insurance to actuarially fair and full insurance e. averse; actuarially fair and full insurance to no insurance
Can you explain how Constant Relative Risk Aversion utility function should be understood and how it works mathematically