Suppose a society contains two individuals. Joe, who smokes, and Tanya, who does not. They each have the same utility function U(C)=ln(C).If they are healthy, they will each get to consume their income of $15,000. If they need medical attention, they will have to spend $10,000, leaving them $5,000 for consumption. Smokers have a 12% chance of needing medical attention, and non smokers have a 2% chance.

An insurance company is willing to insure Joe and Tanya.The twist here is that the insurance company offers two different kinds of policies.One policy is called the low deductible, (L) for which the insurance company will pay any medical costs over $3,000.The other is a high deductible, (H) for which the insurance company will pay any medical costs over $8000.

What is the actuarially fair premium for each type of policy for Joe and Tanya?

If the insurance company can determine who smokes and who does not, and they charge the actuarially fair prices to each, what policy will Joe select?  Tanya?  (Think carefully about calculating expected utilities for each under the different policies.)

Now, suppose that the insurer cannot determine who smokes and who doesn’t.  The insurer sets prices for each product.  The price of L is $840 and the price of H is $40. (Why did I choose these numbers?) What will Joe and Tanya choose to do?  Will adverse selection push Tanya out of the market? [Hint: No.] Calculate the total expected utility for our society under this outcome.

What has happened here?  What does the second policy option accomplish?

E. Suppose the government were to intervene and provide full insurance at a single price and charge everyone the same actuarially fair amount.  How would the total social utility compare to that of part c?  (Ignore any moral hazard or other unintended consequences.)