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In a group of people 10% are unhealthy but don't know it. Of these 50% will likely need to be hospitalized. Of the 90% that are healthy, 10% will likely need be hospitalized. The cost of hospitalization is $50000.

What price of insurance should be offered in the market given:

  1. Neither patients nor the insurance company know who is healthy or un-healthy.

  2. Both patients and the insurance company know who is healthy or not healthy.

  3. Patients know who is healthy or un-healthy but the insurers don't.

I think the answer to 1. is

(0.1 x 0.5 x 50000) + (0.9 x .1 x 50000) = 7000

but I am not sure how to get 2. and 3.

Any guidance would be appreciated.

user1884367
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  • I agree with your math on (1). On (2) and (3), healthy people won't buy insurance, so the insurance company can not sell any policies. There are a lot of insurance company scams around such a situation, but the mathematical bottom line is that in both (2) and (3), the situation is untenable for the insurance company. Even in (2), with the insurance company turning down virtually every prospective customer, the insurance company won't be able to meet its fixed costs (e.g. payroll). – user2661923 Nov 14 '21 at 02:17
  • In real life, you see (2) handled by the insurance company scam (against old people) that the hidden fine print says that the death benefit is limited to the return of premiums for the first ($\approx$) 3 years after the policy begins. Unknown to the customer, the policy becomes the opposite of what the customer wanted - a (subtle) annuity. Further, this annuity is sold to the very people who don't figure to be able to profit from it - the old/sick. Surprisingly, this is all legal. Boy, do the insurance companies have a powerful lobby in the US legislature. – user2661923 Nov 14 '21 at 02:20
  • For 2, the insurance company could sell two different priced policies (assuming that's allowed), in order for the expected payout to equal the price (assuming that's what you are supposed to compute, which of course is not what any for profit insurance company would do). For 3, the insurance company would have to price the insurance for sick people, and no healthy people would buy insurance. – Joe Nov 14 '21 at 11:57

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