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# Expected Utility, Relative Risk Aversion

An expected utility maximizing individual has utility of eno-of-period wealth given by

u(W)= W^(1-y)-1, if y is not equal to 1
ln(W), if y=1

1. Show that this individual has constant relative risk aversion and decreasing absolute risk aversion.

2. Consider the special case where y=2. Suppose that this individual is endowed with an initial wealth, Wo but his end of period wealth is subject to random income shock given as follows

\$y, with probably p
\$0, with probability 1-p

where 0<p<1

He can purchase insurance at a cost of \$c to remove the risk of receiving no income. At what level of initial wealth will he be indifferent between taking on the risk of getting no income and buying the insurance that removes the risk?

#### Solution Preview

See the attached file. The symbols and text here may not print correctly. I have tried to be as simple as possible and have written all the steps so that it is easier for you to understand the solution. Thanks

An expected utility maximizing individual has utility of end-of-period wealth given by

u(W)= W^(1-y)-1, if y is not equal to 1
ln(W), if y=1
1. Show that this individual has constant relative risk aversion and decreasing absolute risk aversion.
u'(W)=(1-y)*W^(1-y-1)= (1-y)*W^(-y) if y is not equal to 1
=W^-1 if y=1
u"(W)=(1-y)*(-y)*W^(-y-1) if y is not equal to 1
=-W^-2 if y=1
Coefficient of Relative Risk Aversion = -W*u"(W)/u'(W)
When y is not equal to 1 we ...

#### Solution Summary

Expected Utility and Relative Risk Aversion are assessed.

\$2.19