$(25$ points$)$ Mr$.$ Fogg is planning an around$-$the$-$world trip. The utility from the trip

is a function of how much he spends on it $(Y)$ given by

$U(Y)=lnY.$

Mr$.$ Fogg has $$10,000$ to spend on the trip.

$($a$)$ If there is a $50\%$ probability that Mr$.$ Fogg will lose $$2,000$ on the trip, what is

his expected utility of the trip? What is his certainty equivalent of his uncertain

income stream of his trip?

$($b$)$ What is the acturarially fair premium to insure his $$2,000$ loss? What is his

maximum willingness to pay to insure his loss?

$($c$)$ To combat the moral hazard problem, the insurance firm requires a $$500$ de$-$

ductible $($i$.$e$.,$ if loss occurs, Mr$.$ Fogg will only get $$1,500$ back$),$ what will be Mr$.$

Fogg's maximum willingness to pay for this insurance policy with the deductible?