6. (15) A individual, with a utility function InW and an initial wealth level Wo, is faced with a fair gamble of winning or losing Sh (where Wo > h> 0) with 50-50 chance. (a) Is this individual risk averse? Explain. (b) Suppose that the individual is willing to pay up to an amount of f in order to avoid such a gamble. Give the equation that determines f, and solve the equation for f (i.e., express fin terms of Wo and h). (c) Show that fincreases as h increases.

A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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**Question 6:** 

An individual, with a utility function \( \ln W \) and an initial wealth level \( W_0 \), is faced with a fair gamble of winning or losing $h (where \( W_0 > h > 0 \)) with a 50-50 chance.

**(a)** Is this individual risk averse? Explain.

**(b)** Suppose that the individual is willing to pay up to an amount of \( f \) in order to avoid such a gamble. Give the equation that determines \( f \), and solve the equation for \( f \) (i.e., express \( f \) in terms of \( W_0 \) and \( h \)).

**(c)** Show that \( f \) increases as \( h \) increases.
Transcribed Image Text:**Question 6:** An individual, with a utility function \( \ln W \) and an initial wealth level \( W_0 \), is faced with a fair gamble of winning or losing $h (where \( W_0 > h > 0 \)) with a 50-50 chance. **(a)** Is this individual risk averse? Explain. **(b)** Suppose that the individual is willing to pay up to an amount of \( f \) in order to avoid such a gamble. Give the equation that determines \( f \), and solve the equation for \( f \) (i.e., express \( f \) in terms of \( W_0 \) and \( h \)). **(c)** Show that \( f \) increases as \( h \) increases.
Expert Solution
Step 1

a) An individual who always refuses fair bets is said to be risk averse. If individuals
exhibit a diminishing marginal utility of wealth, they will be risk averse. As a consequence, they will
be willing to pay something to avoid taking fair bets. To illustrate the connection between risk aversion and insurance, consider a person with a
current wealth of $100,000 who faces the prospect of a 25 percent chance of losing his or her
$20,000 automobile through theft during the next year. Suppose also that this person’s von
Neumann–Morgenstern utility function is logarithmic; that is, U(W ) ¼ ln (W ).
If this person faces next year without insurance, expected utility will be EU ( no insurance) = 0.75U (100, 000) + 0.25U (80, 000)

0.75ln 100,000 + 0.25ln 80,000= 11.45714

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