Question

This is a highly stylized example of an insurance market. Suppose there are just two risk types, A and B, and their probabilities of different levels of medical spending are as shown in the table below (similar to Table 7.1 on page 89 of Reinhardt's book). In this example, we will not be concerned about moral hazard; probabilities and medical spending levels are the same whether or not one has insurance.

Table 1. Probabilities of Different Levels of Medical Spending

Size of Medical bill

Fraction of Type A Fraction of Type B

Expected to Incur Expected to Incur

Bill Such a Bill Such a Bill

$0 0.851 0.5295

$5,000 0.12 0.34

$25,000 0.02 0.09

$100,000 0.009 0.0405

Use this additional information for your answer:

• 80 percent of the population is Risk Type A, 20 percent is Risk Type B
• Individuals know whether they are Type A or Type B, and they know the probabilities of different levels of medical spending (this may be the most unrealistic assumption)
• Insurance is purchased in an individual market
• In order to cover their costs, insurers need to include a loading factor, L = .2, in the premiums they charge
• Individuals are risk-averse up to a point. They are willing to pay up to 1.25 times their "expected" level of medical spending (probability-weighted average) for insurance, but they will not buy insurance if the premium is larger than that amount

Question

1. Suppose first that insurers cannot tell who is Type A and who is Type B, so they must offer the same premium to anyone. If an insurance company could attract a representative population (80 percent Type A , 20 percent Type B), what is the lowest premium it could charge and still cover its costs? (Please show work)
2. What would happen if an insurance company offered a policy at the premium you found in Part 1? Would Type A's buy it? Would Type B's buy it? Would the insurance company cover its costs?