Answered: Show that an agent with utility…

Show that an agent with utility function u(x) = log x is more risk averse than an agent with utility function ˜u(x) = √ x.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.3P

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Concept explainers

Form Utility

The term utility in general refers to the satisfaction earned from usage of a particular product or service. Utility always plays an important role because it influences various important elements like demand & thereby the price of the good or service as well. Though utility is not easy to measure, various types of economic models can be used indirectly. There can be various types of utilities like economic utility, marginal utility, cardinal utility, form utility.

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Show that an agent with utility function u(x) = log x is more risk averse
than an agent with utility function ˜u(x) = √
x.

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Consider the following utility functions: U1(x) = e*; U2(x) = x°, where r > 0 and BE (0,1); U3(x) = 2x + 10. For each function decide whether it belongs to a risk- neutral, risk-averse or risk-loving decision-maker.
Please draw a utility function that exhibits risk-loving behavior for small gambles (low values)and risk-averse behavior for larger gambles (high value).
For the utility function U = Wa, what values of “a” correspond to being risk averse, risk neutral, and risk preferring?
If the utility function is U (W) = ((W0.75) / (0.75)), what is the absolute risk aversion coefficient?
How do you solve this problem? (see attachment)
In the field of financial management, it has been observed that there is a trade-off between the rate of return that one earns on investments and the amount of risk that one must bear to earn that return. a) Draw a set of indifference curves between risk and return for a person that is risk-averse (a person that does not like risk).
6.
Draw a utility function over income u( I) that describes a man who is a risk lover when his income is low but risk averse when his income is high. Can you explain why such a utility function might reasonably describe a person’s preferences?
2- Show which of the following utility functions exhibit decreasing risk aversion: a- U (W) = (W + a)³, a≥0, 00 - d- U(W) = W³
Amy likes to go fast in her new Mustang GT. Their utility function over wealth is v(w) where w is wealth. If Amy goes fast she gets an increase in utility equal to F. But when Amy drives fast, she is more likely to crash: when she drives fast the probability of a crash is 10%, but when she obeys the speed limit, the probability of a crash is only 5%. Amy's car is worth $2000 unless she crashes, in which case it is worth $0. If Amy doesn't have insurance, driving fast isn't worth the risk, so she will alway obey the speed limit. If Amy is offered an insurance contract with full insurance for a premium P with the deductible D, which of the inequalites below is her incentive compatibility constraint that makes sure that she will still obey the speed limit even when she is fully insured? 0.05U(2000 – P – D) + 0.95U(2000 – P) > 0.05U(0 – P – D + 2000) + 0.95U(2000 – P) 0.05U(2000 – P – D) + 0.95U(2000 – P) > 0.1(U(2000 – P – D) + F) + 0.90(U(2000 – P) + F) 0.05U(2000 – P – D) + 0.95U(2000)…
A driver's wealth $100,000 includes a car of $20,000. To install a car alarm costs the driver $1,750. The probability that the car is stolen is 0.2 when the car does not have an alarm and 0.1 when the car does have an alarm. Assume the driver's von Neumann- Morgenstern utility function is U(W) = In(W). Suppose the driver is deciding between the following three options: (a) purchase no car insurance, do not install car alarm; (b) purchase fair insurance to replace the car, do not install car alarm; and (c) purchase no car insurance, install car alarm. Of these three options, the driver prefers: A. option (a). B. option (b). C.option (c). D.options (a) and (b). E. options (a) and (c). F. options (b) and (c). G.all options equally. H.none of these options.
Can you explain how Constant Relative Risk Aversion utility function should be understood and how it works mathematically