Problem $2.$ This problem takes you through the formal definition of risk aversion $/$ risk

loving. Given a lottery P $,$ let E $($P $)$ be the expected value of the lottery P $.$ For example, if

P $=($$$10,0\mathrm{.}5$; $$0,0\mathrm{.}5),$ then

E $($P $)=0\mathrm{.}5\backslash$times $10+0\mathrm{.}5\backslash$times $0=5$

We say a person is

$$ Risk averse if he chooses E $($P $)$ dollars for sure over the lottery P

$$ Risk neutral if he is indifferent between E $($P $)$ dollars and the lottery P

$$ Risk loving if he chooses the lottery P over E $($P $)$ dollars for sure.

$(1)$ Ann has vNM utility u$1($x$)=$ x$,$ Bob has utility u$2($x$)=$ log $($x $+1)$ and Carl has

utility u$3($x$)=$ x$2.$ Who is risk neutral, risk averse and risk loving? $(5$ points$)$

$(2)$ Consider the lottery P again. Find the dollar amount x such that each person is

indifferent between the lottery P and $x $($x is the certainty equivalent of P $)(10$

points$)$

One way to measure $$how$$ risk averse someone is is to use something called the Arrow$-$Pratt

coefficient of risk aversion. Given a vNM utility u$,$ the Arrow$-$Pratt coefficient is

r $($x$)=$u$($x$)$

u$($x$)$

$(3)$ Calculate the Arrow$-$Pratt coefficients for everyone. How do they compare? Does this

agree with your answers before? $(10$ points$)$

$(4)$ Calculate the Arrow$-$Pratt coefficient for utility u $($x$)=$e$\backslash$rho x where $\backslash$rho $>0.$ This type

of utility is called Constant Absolute Risk Aversion $($CARA$).$ Why do you think

it$$s called CARA? $(5$ points$)$

$(5)$ Calculate the Arrow$-$Pratt coefficient for utility u $($x$)=$ x$1\backslash$rho

$1\backslash$rho where $\backslash$rho $>0.$ This type

of utility is called Constant Relative Risk Aversion $($CRRA$).$ Why do you think

it$$s called CRRA? $(5$ points$)$

$2$