Problem 4: Recall the following portfolio allocation problem: an individualwith initial wealth wo has to choose an allocation between a safe asset (with azero rate of return) and a risky asset with a random rate of return & € [−1, 1],where E[8] > 0. If the individual invests & dollars in the risky asset, then finalwealth is (wo-x)+x(1+5) = w₁ +x6. The individual chooses to maximizeexpected utility denoted by E[u(wo +xd)].1. Suppose the Bernoulli utility function is u(w)solute risk aversion is a constant.= -e. Show that ab-2. Let x* (wo) denote the optimal amount of investment in the risky asset.Show that for the utility function of Part a., da = 0, i.e. the amount ofinvestment in the risky asset is independent of initial wealth. Explain thisresult.dwo3. Now suppose that the utility function is u (w) = aw - bw², where a, b > 0.The parameters a and b are such that marginal utility of wealth is positivefor all w. What can we say about in this case? Explain your result.

The utility function of the consumer shows the satisfaction received by the consumer is based on the…

Problem 4: Recall the following portfolio allocation problem: an individual with initial wealth wo has to choose an allocation between a safe asset (with a zero rate of return) and a risky asset with a random rate of return = [−1,1], where E[6] > 0. If the individual invests x dollars in the risky asset, then final wealth is (wo-x) + x (1 + d) = w₁ +xd. The individual chooses x to maximize expected utility denoted by E[u(w₁ + xd)]. 1. Suppose the Bernoulli utility function is u(w) solute risk aversion is a constant. dwo ew. Show that ab- 2. Let x*(wo) denote the optimal amount of investment in the risky asset. Show that for the utility function of Part a., da = 0, i.e. the amount of investment in the risky asset is independent of initial wealth. Explain this result. 3. Now suppose that the utility function is u (w) = aw – ½bw², where a, b > 0. The parameters a and b are such that marginal utility of wealth is positive for all w. What can we say about in this case? Explain your result.

Problem 4: Recall the following portfolio allocation problem: an individual with initial wealth wo has to choose an allocation between a safe asset (with a zero rate of return) and a risky asset with a random rate of return = [−1,1], where E[6] > 0. If the individual invests x dollars in the risky asset, then final wealth is (wo-x) + x (1 + d) = w₁ +xd. The individual chooses x to maximize expected utility denoted by E[u(w₁ + xd)]. 1. Suppose the Bernoulli utility function is u(w) solute risk aversion is a constant. dwo ew. Show that ab- 2. Let x*(wo) denote the optimal amount of investment in the risky asset. Show that for the utility function of Part a., da = 0, i.e. the amount of investment in the risky asset is independent of initial wealth. Explain this result. 3. Now suppose that the utility function is u (w) = aw – ½bw², where a, b > 0. The parameters a and b are such that marginal utility of wealth is positive for all w. What can we say about in this case? Explain your result.

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

Problem 4: Recall the following portfolio allocation problem: an individual
with initial wealth wo has to choose an allocation between a safe asset (with a
zero rate of return) and a risky asset with a random rate of return & € [−1, 1],
where E[8] > 0. If the individual invests & dollars in the risky asset, then final
wealth is (wo-x)+x(1+5) = w₁ +x6. The individual chooses to maximize
expected utility denoted by E[u(wo +xd)].
1. Suppose the Bernoulli utility function is u(w)
solute risk aversion is a constant.
= -e. Show that ab-
2. Let x* (wo) denote the optimal amount of investment in the risky asset.
Show that for the utility function of Part a., da = 0, i.e. the amount of
investment in the risky asset is independent of initial wealth. Explain this
result.
dwo
3. Now suppose that the utility function is u (w) = aw - bw², where a, b > 0.
The parameters a and b are such that marginal utility of wealth is positive
for all w. What can we say about in this case? Explain your result.

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