1. (i) A utility function has the form U(w) = aw³ +bw2 where a, b E R. Assuming w > 0 and a 0, is U(w) ever suitable for a non-satiated and risk-averse investor? If it is, state the range of values of w this holds for as a function of a and b. You must include your working. (ii) Give an example of a utility function with DARA. Show all your working. (iii) An investor has IRRA. Would this investor prefer a fair gamble or to do nothing? Provide a short explanation of your answer.

1. (i) A utility function has the form U(w) = aw³ +bw2 where a, b E R. Assuming w > 0 and a 0, is U(w) ever suitable for a non-satiated and risk-averse investor? If it is, state the range of values of w this holds for as a function of a and b. You must include your working. (ii) Give an example of a utility function with DARA. Show all your working. (iii) An investor has IRRA. Would this investor prefer a fair gamble or to do nothing? Provide a short explanation of your answer.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.8P

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a) Compute the (absolute) risk aversion measure dependent r(W) of utility function -e -aW Is r(W) on W?
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2 (a) Consider the St. Petersburg Paradox problem first discussed by Daniel Bernoulli in 1738. The game consists of tossing a coin. The player gets a payoff of 2" where n is the number of times the coin is tossed to get the first head. So, if the sequence of tosses yields TTTH, you get a payoff of 2¹; this payoff occurs with probability. Compute the expected value of playing this game. (b) Assume that utility U is a function of wealth X given by U = X5 and that X= $1,000,000. In this part of the question, assume that the game ends if the first head has not occurred after 40 tosses of the coin. In that case, the payoff is 240 and the game is over. What is the expected payout of this game? (c) What is the most you would pay to play the game if you require that your expected utility after playing the game must be equal to your utility before playing the game? Use the Goal Seek function (found in Data, What-If Analysis) in Excel.
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5. Shift-in-charge Nazar Al Rushdy: Nazar is pessimistic about the market price. What is your guidance for Nazar? The decision to employ decision trees in crucial situations has been taken by Salem Al Harthi, the plant manager. The table below presents data on demand for a duration of 6 hours along with their respective probabilities. The first row of the table provides the probability of demand for the initial three hours when a leak occurs, denoted in parentheses. Subsequently, the following three rows indicate the probabilities of high, medium, and low demand for the succeeding three hours. To illustrate, if the initial 3-hour market price was low, the probabilities of high demand, medium demand, and low demand in the next three hours are 0.2, 0.3, and 0.5, respectively. Market price High Market price Medium Initial 3-hrs (0.2) Initial 3-hrs (0.5) Market price Low Initial 3-hrs (0.3) High demand (next (0.5) (0.4) (0.2) 3 hrs) Medium demand (0.3) (0.2) (0.3) (next 3 hours) Low demand…
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Q1. A farmer believes there is a 50-50 chance that the next growing season will be abnormally rainy. His expected utility function has the form Expected utility = 0.5lnYNR + 0.5lnYR Where and represent the farmers income in the state of ‘normal rain’ and ‘rainy’ respectively. Suppose the farmer must choose between two crops that promise the following income prospects Crop YNR YR Wheat $83,000 $10,000 Maize $83,000 $15000 What mix of wheat and maize would provide maximum expected utility to this farmer?
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