Let $U(W)$ denote the utility of wealth $(W),$ such that

$]=[0$

$($a$)$ Sketch the utility function in $(7)$ for $a=-1,a=0$ and $a=1.$

$($b$)$ How can we characterize the behaviour of individuals with $a=-1,a=0$ and $a=1?$ What are the implications for the relation between expected utility of wealth and utility of expected wealth for uncertain levels of future wealth?

$($c$)$ Calculate the absolute and relative risk aversion coefficient for $a=-1,a=0$ and $a=1.$ Does any of these parametrizations describe a realistic behaviour of an individual? Explain.

$($d$)$ For $A>0,$ show that if wealth $W$ is normally distributed with mean $$ and variance ${}^2,$ then expected utility can be expressed as a simple linear equation in $$ and ${}^2.$