Consider an individual who has an isoelastic utility function with $\backslash (\backslash$gamma$=1\backslash )$ and $\backslash (\backslash$$ $50,000\backslash )$ in liquid financial capital $($wealth$).$ They are purchasing insurance for their car and are unsure about the size of the deductible. The current value of car is $\backslash (\backslash$$ $35,000\backslash ),$ and for the sake of simplicity, assume the only kind of accident they can get in results in their car being completely destroyed. Further suppose that the probability this happens is equal to $\backslash (0\mathrm{.}2\backslash \%\backslash )$ Finally, suppose they are choosing between the following three insurance policies $($I repeat some information to help clarify$)$:

Policy $1$

$-$ Probability of loss $\backslash (=0\mathrm{.}2\backslash \%\backslash ).$

$-$ Loss $\backslash (=\backslash$$ $35,000\backslash ).$

$-$ Remember, we are assuming they can only purchase full coverage. le$.,$ they are purchasing $35,000$ in coverage.

$-$ Deductible $\backslash (=\backslash$$ $0\backslash )($i$.$e$.,$ no deductible$).$

$-$ Premium $\backslash (=\backslash$$ $105\backslash ).$

Policy $2$

$-$ Probability of loss $\backslash (=0\mathrm{.}2\backslash \%\backslash ).$

$-$ Loss $\backslash (=\backslash$$ $35,000\backslash ).$

$-$ For simplicity, assume the estimated probability of loss does not change across insurance providers.

$-$ Deductible $\backslash (=\backslash$$ $500\backslash ).$

$-$ Premium $\backslash (=\backslash$$ $100\backslash ).$

Policy $3$

$-$ Probability of loss $\backslash (=0\mathrm{.}2\backslash \%\backslash ).$

$-$ Loss $\backslash (=\backslash$$ $35,000\backslash ).$

$-$ Deductible $\backslash (=\backslash$$ $3000\backslash ).$

$-$ Premium $\backslash (=\backslash$$ $85\backslash ).$

Part A $(4$ marks$)$: Please compute, and comment on$,$ the individual's reservation price for Policy $1.$

Part B $(6$ marks$)$: Which insurance policy does the individual choose? $($Hint: you can do this without computing the reservation prices$).$

Part C $(3$ marks$)$: In general, how will the insurance premium $($price charged by the insurance provider$)$ vary with the size of the deductible, holding all else constant? Note: This is not a trick question and should be $($hopefully is$)$ intuitive.

Part D $(4$ marks$)$: Explain the concept of diminishing marginal utility in the context of insurance. Put differently, what is the effect of diminishing marginal utility on the reservation price?

Part E $(3$ marks$)$: In words, explain how the reservation price changes with wealth for someone whose preferences exhibit increasing marginal utility.

Reminder: marginal utility simply refers to the first$-$order derivative of utility with respect to wealth.