Question

Consider the market for health insurance. Suppose there are three people. Healthy Hal, Average Al,
and Portly Pete. All three of these people will buy health insurance if it costs less than or equal to his
actuarily fair insurance rate. Hal's true cost to insure is $5,000, Al's true cost to insure is $10,000, and
Pete's true cost is $15,000.
a) Suppose the insurance company must offer a single policy to all customers, and it tries to be fair by
averaging the costs of its three customers. What premium will it charge? Who will choose to buy
insurance at that price?
b) Observing who has actually purchased a policy, the insurance company decides to adjust its
premium to be the average cost of the observed customers in part (a). What premium will it charge?
Which customers will buy insurance at this price?
c) Once again, the insurance company observes who actually purchased insurance, and decides to
adjust its premium to the average cost of its observed customers in part (b). What premium will
charge? Which customers will buy insurance at this price? Is this market efficient? If not, suggest
some mechanism that might improve its efficiency.

For part (a), 1/3(5000+10000+15000)=10000? Al and Pete will buy
For part (b), 1/2(10000+15000)=125000; Pete will buy. The market is not efficient.
For part (c), 15000; Pete will buy
Am I right for the answers? And what will be the mechanisms to improve the market?
Thanks.

Answer 1

Consider the market for health insurance. Suppose there are three people. Healthy Hal, Average Al, and Portly Pete. All three of these people will buy health insurance if it costs less than or equal to his actuarily fair insurance rate. Hal's true cost to insure is $5,000, Al's true cost to insure is $10,000, and Pete's true cost is $15,000.

a) Suppose the insurance company must offer a single policy to all customers, and it tries to be fair by averaging the costs of its three customers. What premium will it charge? Who will choose to buy insurance at that price?

You wrote:
(a), 1/3(5000+10000+15000)=10000? Al and Pete will buy

We're told that all three will buy health insurance if it is ≤ his actuarily fair insurance rate

So you are correct! Hal won't buy because 10,000 is NOT less than or equal to 5,000.

b) Observing who has actually purchased a policy, the insurance company decides to adjust its premium to be the average cost of the observed customers in part (a). What premium will it charge? Which customers will buy insurance at this price?

You wrote:

For part (b), 1/2(10000+15000)=125000; Pete will buy. The market is not efficient

Yes! However, it will be $12,500, not $125,000 - I think you added an extra zero in there!

Pete will buy insurance at this price but the others won't. Moreover, the market is not efficient - yep!

c) Once again, the insurance company observes who actually purchased insurance, and decides to adjust its premium to the average cost of its observed customers in part (b). What premium will charge? Which customers will buy insurance at this price? Is this market efficient? If not, suggest some mechanism that might improve its efficiency.

You wrote:

. For part (c), 15000; Pete will buy. Correct.

what will be the mechanisms to improve the market? Great question. Some mechanisms:

a) single-payer health care system

b) universal (i.e. tax-based insurance - the graduated income tax will take care of it)

I hope that you found this answer useful towards your studies. It took a considerable amount of thought, time, and effort to compose, and and I'd sincerely appreciate a lifesaver rating! It would really make my day! :)