Question

There are 10000 demanders of health insurance in Pittsburgh, PA. They each have wealth of $10 (in thousands) and their utility of wealth is U(W) = W1/2. An insurance company interested in selling to them knows only that half of Pittsburgh’s residents are of weak constitutions and with a 60% probability will require $4 (in thousands) of medical care in the coming year, and that the other half are of strong constitutions and will require $4 of (in thousands) care with a 40% probability. The company cannot tell which are weak (W) and which are strong (S).

a.If the insurance company could sell policies to all 10000 residents, what premium would allow it to break even?

b.If the insurance company charges the premium found in part a., will all of the residents of Pittsburgh buy the insurance?

c.What profit can the company expect to earn? Will the insurance company charge the premium found in part a.? If not, what will the fair premium be?

Answer 1

a. To solve for break-even premium find the expected value of the pay-out by insurance company per resident.

Total residents = 10,000.

Half of total residents = 5,000

Expected pay out for a resident. with weak constitution = probability of medical care x cost of medical care = 0.6 x $4,000 = $2,400 per weak resident. Or, total payout for residents with weak consitition = $2,400 x 5,000 = $12,000,000 ($12 million).

Expected payout for a resident with strong constitution = probability of medical care x cost of medical care = 0.4 x$4,000 = $1,600. Or, total payout for residents with strong consitition = $1,600x5000 = $8,000,000 ($8 million).

Total expected pay-out = $12 million +$8 million = $20 million.

Expected payout per resident = $20,000,000 / 10,000 = $2000.

If the insurance company charges a premium equal to its expected pay-out per resident it will break-even. Therefore the break-even premium is $2,000.

b.

If insurance company charges premium in part a, those with expected value above (week constitution) it will buy insurance and those with expected value value below it (strong constitution) will not buy insurance, assuming the residents know whether they have a weak or a strong constitution.

c.

Company can not expect to earn a profit, because its expected pay-out is more than what it gets as a premium.

Insurance company will not charge the premium found it part a.

Its fair premium will be $2,400, the expected payout for weak constitution residents.