A Deductible Is an Amount of a Claim Not Covered

Essay

A deductible is an amount of a claim not covered by insurance.A deductible is a fixed portion of the accident cost that the insured person must pay in order to make a claim to their insurer (this is similar to a co-pay in which you must pay a portion of the medical costs in the case you get sick).For example,if I break my arm,I have to pay a $50 deductible to the insurance company in order to get them to cover the rest of my medical costs from the accident.
Rosa has a 10% chance of getting sick in the next year.If she gets sick,her medical bills will amount to $500.She has a wealth of $1,000.Suppose she has the utility function U(X)= X^0.5 where x is her net wealth at the end of the year.
Suppose Rosa can purchase insurance.The insurance company provides two plans for Rosa to select from,both providing $500 of coverage in the case that Rosa gets hurt.Plan A has zero deductibles (good!)but charges a high premium (bad!).Specifically,Plan A charges $55 for $500 of coverage.Plan B has a deductible of $K,where K < 500,but charges a premium of just $(55 - .11K).
a.Suppose K = $100.Will Rosa purchase insurance and if so,which plan? Show this mathematically.
b.If Rosa can choose the deductible,K,what amount of deductible will she choose?
c.Suppose Rosa knows her chance of getting sick is really just 5%.How will this affect the deductible she chooses? You only need to provide intuition (in words)for this part,not explicit computation.