Question

1. Suppose an individual considering purchasing health insurance has a 10% chance of developing a condition that costs $50,000 to treat. The individual has initial disposable income of $90,000 and is risk averse, with utility over final disposable income given by v ( I ) = sq root I.

a) What is the highest premium this individual is willing to pay for an insurance policy that reimburses them $50,000 if they need treatment? (20 points)

b) Now suppose that indiviudals may be either low risk (in which case the probability of developing the condition is 10%) or high risk (in which case the probability of developing the condition is 90%). If the insurance provider cannot tell which category a given individual falls into, what is a fair premium for a policy that pays out $50,000 if treatment is needed? (5 points)

c) Now suppose that although the insurance company cannot tell the individuals risk category, the individuals purchasing insurance know their own risk category. If the insurance company offers two policies: one which pays out $50,000 in the event that treatment is needed, with a premium of $25,000 and another which pays out $2,500 if treatment is needed, with a premium of $500, which policy would be chosen by high risk individuals and which policy would be chosen by low risk individuals? (10 points)

Answer 1

a) Disposable income when not ill =90000 with utility =√90000=300

Disposable income when ill=90000-50000=40000 with utility=√40000=200

Probable utility =0.9*300(probability not ill* utility not ill)+0.1*200(probability ill * Utility ill)=290

Which will correspond to disposable income "I" given as √I=290 or I=84100 Hence ,Maximum premium individual can pay is initial income -84100=90000-84100=5900

b) Fair premium will be expected treatment cost where as probability for high risk and low risk is 50% each so expected cost for low risk =50%*10%*50000=2500

And for High risk =50%*90%*50000=22500,.

So fair premium =2500+22500=25000

c) For a payment of 50000 with 25000 premium utility with insurance for high cost=√(90000-25000(cost of premium)=√65000=254.95 where as utility without insurance =0.9*√(90000-50000)+0.1√(90000)=180+30=210 Hence utility is greater with insurance so High risk person will buy insurance with premium of 25000 for 50000 payment

Now, for low risk at 2500 payment for 500 premium utility with insurance=√(90000-500)=299.165 and Utility without insurance=0.1*√40000+0.9*√90000=20+270=290 Since utility is higher with insurance Low risk person will buy 2500 payment at premium of 500