Exercise $2$: Uhcertainty and I nsurance. $(50$ pts$)$

You are an owner of a luxurious sailing boat, worth $$10$ that you use for recreation on

Mansfield lake. Unfortunately, there is a good $(50\%)$ chance of a tornado in Storrs $($probability

is equal to $\frac{1}{2})$ that completely destroys it$.$ Thus, your boat is in fact a lottery with payment

$(0,10).$

$($a$)$ What is the expected value of the "boat lottery"? $($Give one number.$)$

$($b$)$ Suppose your Bernoulli utility function is given by $u(c)=c^2.$ Give von Neuman $-$

Morgensternutility functionover lotteries $U(C_1,C_2)($theformula$).$

$($c$)$ Your Bernoulli utility function changes to $u(c)=lnc.$ Give the von Neuman$-$

Morgenstern utility function. $($A formula.$)$

$($d$)$ You can insure your boat by buying insurance policy in which you specify coverage

$x.$ The insurance contract costs $x$ and the premium rate is $=\frac{1}{2}.$ Find analytically

and depict in the graph your budget constraint. Mark the point that corresponds to

no insurance.

$($e$)$ Find optimal level of coverage $x.$ Are you going to fully insure your boat? $($Give

one number and yes$/$no answer.$)$ Depict the optimal consumption plan on the graph.

$($f$)$ Propose a premium rate for which you will only partially insure your boat. $($Give

one number.$)$

Please make sure to include graph