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Consider the following utility function of an agent (note that x can be understood as each possib...
(1) Ann has vNM utility u1 (x) = x, Bob has utility u2 (x) = √...
(1) Ann has vNM utility u1 (x) = x, Bob has utility u2 (x) = √ x and Carl has utility u3 (x) = x 3 . Who is risk neutral, risk averse and risk loving? (2) Consider the lottery P again. Find the dollar amount x such that each person is indifferent between the lottery P and $x (x is the certainty equivalent of P) (3) Calculate the Arrow-Pratt coefficients for everyone. How do they compare? Does this agree...
Consider the utility function u(x) = ax + b e^cx where a, b, c are positive...
Consider the utility function u(x) = ax + b e^cx where a, b, c are positive scalars. (a) Compute the coefficient of absolute risk aversion. (b) Describe the risk attitude represented by u(x) and how it changes as x increases. (c) Write down the equations to determine the certainty equivalent and the risk premium of a gamble X for an individual with initial wealth w > 0. (d) What is the sign of the risk premium? How does the risk...
W 2 Assume you have the following von Neumann-Morgenstern utility function: U(w) = 200 – (12...
W 2 Assume you have the following von Neumann-Morgenstern utility function: U(w) = 200 – (12 – 1000) where w denotes wealth measured in pounds. Your friend believes that it will rain in Reading tomorrow while you think that there is a 75% probability that it will not rain. You are going to take a bet. Your friend will pay you x pounds if it does not rain and you pay your friend x pounds if it rains. Your current...
Question 8: Consider a decision-maker with utility function u(x) = x^0.8 , where x>0 denotes the...
Question 8: Consider a decision-maker with utility function u(x) = x^0.8 , where x>0 denotes the decision-maker’s wealth. a. Determine the decision-maker’s attitude towards risk. In other words, is this decision-maker risk-neutral, risk-averse or risk-loving? Provide a justification of your answer. Solution: We have ?''(x) = -0.16x^-1.2 <0 . Hence, the decision-maker is risk-averse. Please explain solution. How did he get the answer?
9. Problems and Applications 29 Dmitri has a utility function U =wi, where W is his...
9. Problems and Applications 29 Dmitri has a utility function U =wi, where W is his wealth in millions of dollars and U is the utility he obtains from that wealth. In the final stage of a game show, the host offers Dmitri a choice between (A) $4 million for sure, or (B) a gamble that pays $1 million with probability 0.4 and $9 million with probability 0.6. Use the blue curve (circle points) to graph Dmitri's utility function at...
Consider the data in the table below and answer the following questions: Utility Score Portfolio L...
Consider the data in the table below and answer the following questions: Utility Score Portfolio L Utility Score Portfolio M Utility Score Portfolio H Investor Risk Aversion (A) Er) =.07: =.05 E(r)=.09: O= E(r)= 13: o = 2 13-4x2x.22 =.0900 107 _x2x.052 = .0675.09–5x2x. P = 0800 <3<.05º =.0663.00 – £x3x8 =.0750 13-_x3x.2° = 0700 2X4x.052 - 0650.09 -->x4x. 1° = -0700 13x4x.22 - 0500 1. The three risk aversion coefficients in the first column represent investors X, Y and...
12. Which of the following statements is necessarily false for some given risky prospect, X (assume...
12. Which of the following statements is necessarily false for some given risky prospect, X (assume that X has more than one possible outcome)? a. An individual with convex utility of wealth will have a greater value of CE(X) than an individual with concave utility of wealth? b. When utility of wealth is linear, the numeric value (ignoring units) of CE(X) - EV(X) is equivalent to the numeric value of EU(X) - U(EV(X)). c. A doubling of all of X's...
Suppose that a decision maker’s utility as a function of her wealth, x, is given by...
Suppose that a decision maker’s utility as a function of her wealth, x, is given by U(x) = ln (2x) (where ln is the natural logarithm). The decision maker now has $10,000 and two possible decisions. For Alternative 1, she loses $1000 for certain. For Alternative 2, she gains $500 with probability 0.8 and loses $2,000 with probability 0.2. Which alternative should she choose and what is her expected utility (rounded to 2 decimals)? a.She should choose Alternative 1. Her...
1. Consider an agent who values consumption in period 0 and 1 according to the following...
1. Consider an agent who values consumption in period 0 and 1 according to the following utility function: u(co, C)In(Co)+8 In(c1) is a discount factor (5 < 1) which indicates that the agnet prefers to consume today more than he can tomorrow. Suppose that the agent is given a total wealth today of w and that he may save any portion of this money in order to consume tomorrow. If he saves money he is paid interest r. Thus the...
Problem 1 Consider the following two-period utility maximization problem. This utility function belongs to the CRRA...
Problem 1 Consider the following two-period utility maximization problem. This utility function belongs to the CRRA (Constant Relative Risk Aversion) class of functions which can be thought of as generalized logarithmic functions. An agent lives for two periods and in both receives some positive income. subject to +6+1 4+1 = 3+1 + (1 + r) ar+1 where a > 0,13 € (0, 1) and r>-1. (a) Rewrite the budget constraints into a single lifetime budget constraint and set up the...
W 2 Assume you have the following von Neumann-Morgenstern utility function: U(w) = 200 – (12 – 1000) where w denotes wealth measured in pounds. Your friend believes that it will rain in Reading tomorrow while you think that there is a 75% probability that it will not rain. You are going to take a bet. Your friend will pay you x pounds if it does not rain and you pay your friend x pounds if it rains. Your current...
9. Problems and Applications 29 Dmitri has a utility function U =wi, where W is his wealth in millions of dollars and U is the utility he obtains from that wealth. In the final stage of a game show, the host offers Dmitri a choice between (A) $4 million for sure, or (B) a gamble that pays $1 million with probability 0.4 and $9 million with probability 0.6. Use the blue curve (circle points) to graph Dmitri's utility function at...
Consider the data in the table below and answer the following questions: Utility Score Portfolio L Utility Score Portfolio M Utility Score Portfolio H Investor Risk Aversion (A) Er) =.07: =.05 E(r)=.09: O= E(r)= 13: o = 2 13-4x2x.22 =.0900 107 _x2x.052 = .0675.09–5x2x. P = 0800 <3<.05º =.0663.00 – £x3x8 =.0750 13-_x3x.2° = 0700 2X4x.052 - 0650.09 -->x4x. 1° = -0700 13x4x.22 - 0500 1. The three risk aversion coefficients in the first column represent investors X, Y and...
12. Which of the following statements is necessarily false for some given risky prospect, X (assume that X has more than one possible outcome)? a. An individual with convex utility of wealth will have a greater value of CE(X) than an individual with concave utility of wealth? b. When utility of wealth is linear, the numeric value (ignoring units) of CE(X) - EV(X) is equivalent to the numeric value of EU(X) - U(EV(X)). c. A doubling of all of X's...
1. Consider an agent who values consumption in period 0 and 1 according to the following utility function: u(co, C)In(Co)+8 In(c1) is a discount factor (5 < 1) which indicates that the agnet prefers to consume today more than he can tomorrow. Suppose that the agent is given a total wealth today of w and that he may save any portion of this money in order to consume tomorrow. If he saves money he is paid interest r. Thus the...
Problem 1 Consider the following two-period utility maximization problem. This utility function belongs to the CRRA (Constant Relative Risk Aversion) class of functions which can be thought of as generalized logarithmic functions. An agent lives for two periods and in both receives some positive income. subject to +6+1 4+1 = 3+1 + (1 + r) ar+1 where a > 0,13 € (0, 1) and r>-1. (a) Rewrite the budget constraints into a single lifetime budget constraint and set up the...
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