Timah has a utility function u$($w$)=$ ln w and she has $$50000$ as her wealth. However, she lives

in a flood$-$prone area. She expects flood occurs with probability $0\mathrm{.}2$ in any given year. If there

is flood $($F$),$ her house would suffer water damages that costs $$10000.$ If there is no flood $($N$),$

she won$$t have any loss. Suppose for now that there is no insurance available.

$($a$)$ Find her expected wealth.

$($b$)$ Find her expected utility.

$($c$)$ Find the certainty equivalent and the risk premium. Give an economic interpretation to

these numbers.

For the rest of the question, suppose instead that Timah can choose to buy as much in$-$

surance as she wants at a price of p per dollar of insurance. This means that to get q

dollars of compensation when flood occurs, she has to pay pq dollars as insurance premium.

d$)$ Suppose Timah chooses to purchase an insurance that compensates q dollars in case of

a flood. What is her wealth, wF $,$ if there is flood? What is her wealth, wN $,$ if there is

no flood? $($Note that the premium needs to be paid before she knows whether there is a

flood.$)$

$($e$)$ Given the price p$,$ Timah would choose q to maximize her expected utility. Write down

her expected utility maximization problem in math.

$($f$)$ Solve for the first order condition.$($Note: If we consider the wealth in the state of flood

and the wealth in the state of no flood as two consumption goods, with prices equal to p

and $1$ p$,$ this condition can be rearranged to show that M RS equal to the price ratio.$)$

$($g$)$ Suppose that the insurance is not actuarially fair, meaning that p $>0\mathrm{.}2.$ Would she fully

insure against the loss, i$.$e$.$ q $=10000?$ Justify your answer.