There is an unknown virus spreading and making people sick in early 2020. The virus is known to infect only two per cent of the population; only they require hospitalization. Infected people make up a further 10 per cent of the population but are asymptomatic and unaware they are infected. As for the remaining 88 per cent of the population, they are not infected with the disease. As of early 2020, there is still no test for determining whether a person is asymptomatic even if they have the virus. Hospitalization costs are $40,000 on average. Those who are hospitalized in the United States cannot buy health insurance since they already have the condition. In these cases, the unlucky few must cover their own healthcare expenses.

There is a 50 per cent chance that patients with no symptoms will need to be hospitalized. The likelihood that an uninfected patient will have to be hospitalized eventually is 10%. If the virus is contracted, but no hospitalization is required, people recover at home and do not have to use insurance. Asymptomatic and uninfected populations alike want to be covered as soon as possible.

(i) Assuming asymptomatic and infected patients both want insurance, how much would a competitive insurer charge for expected costs?

(Insurance companies and prospective patients do not know who is infected and asymptomatic or uninfected).

(ii) Full information scenario

Imagine a test was available that could definitively determine if an asymptomatic person had the disease. It is also possible to use the test to determine whether a person is uninfected. The test will be paid for by the government, so neither patients nor insurance companies need to pay for it. How much does insurance cost on the market:

The price for asymptomatic infected patients is:

The price for uninfected patients is