Let X be a random variable determining an individual$$s income next year, given her likeliness of being

sick. Let P r$($x$)$ be the probability mass function $($pmf$)$ of this variable. An individual is an expected

utility maximizer, with utility function given by

U $($x$)=$ X

x

u$($x$)$P r$($X $=$ x$)$

u$($x$)=$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$,$

u$($c$)=$ X

x

u$($x$)$P r$($X $=$ x$)$

Moreover, define the Arrow Pratt coefficient of absolute risk aversion at x as rA$($x$)=$u$($x$)/$u$($x$),$ and

the coefficient of relative risk aversion as rR$($x$,$ u$)=$xu$($x$)/$u$($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?

$($c$)$ Calculate the certainty equivalent for when X takes values $(16,4)$ with probabilities $1$

$2,1$

$2$

respec$-$

tively. Is it higher or lower than the expected value of this random variable?

$($d$)$ Calculate the certainty equivalent for when X takes values $(36,16)$ with probabilities $1$

$2,1$

$2$

respec$-$

tively. Compare this result with the one in the previous question and interpret.