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 TypeA 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 TypeA, 20 percent is Risk Type B
• Individuals know whether they are TypeA 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 TypeA 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 TypeA, 20 percent Type B), what is the lowest premium it could charge and still cover its costs? (Please show work)

Step-by-step explanation FOR QUESTION 1

1.Supposing the insurers cannot tell who is TypeA and who is Type B.

Premium = [ probability of Risk Type a (sum of (probability *(its bill value)))] + [probability of risk type B (sum of (probability *(its bill value)))]

= [0.8(0(0.851) + 0.12 (5000) + 0.02 (25000) +0.009 (1,00,000))] + [0.2(0(0.5295) + 0.34(5000) + 0.09(25000) + 0.0405(100,000)]

= 0.8(0 + 600 + 500 + 900) + 0.2(0 + 1,700 +2,250 + 4,050)

= 0.8(2000) + 0.2(8000)

=1600 + 1600

= $3200

In order to cover the cost, insurers need to include a loading factor, L = 0.2, in the premiums they charge.

Therefore, the lowest premium they should charge is: 3200 + 0.2(3200) = $3,840.

****NEED HELP WITH QUESTION 2 AND 3***

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?

3.In light of your answers to Parts 1 and 2, what would you expect equilibrium to look like in this market? Who would buy insurance and at what premium? (You should assume that in equilibrium, competition among insurers keeps premiums as low as possible, consistent with covering costs.)