Let $x$ be a random variable determining an individual's income next year, given her likeliness of being sick. Let $Pr(x)$ be the probability mass f fimpction $($pmf$)$ of this variable. An individual is an expected utility maximizer, with utility function given by

$)=(x$

The certainty equivalent, denoted $c,$ is the amount of money for which the individual is indifferent between the gamble on consuming a random amount given by the random variable $x$ versus the certain amount $c$; that is$,$

$)=(x$

Moreover, define the Arrow Pratt coefficient of absolute risk aversion at $x$ as $r_A(x)=-u^{\text{'}\text{'}}\frac{x}{u^{\text{'}}}(x),$ and the coefficient of relative risk aversion as $r_R(x,u)=-x{u}^{\text{'}\text{'}}\frac{x}{u^{\text{'}}}(x).$

$($a$)$ Calculate the Arrow$-$Pratt coefficients of absolute and relative risk aversion at the level of income $x=5.$

$($b$)$ What is the meaning of these$-$two risk aversion metrics? Why would we use one vs the other?