Suppose there are 4 group of people of differing risk catego

Suppose there are 4 group of people of differing risk categories, with a large and equal number of people in each category. Insurers cannot tell which group a person belongs to, so we have asymmetric information since the individuals know their own risk category. Each group faces a risk of loss of $10,000. Suppose that the willing to pay for insurance of people in each category is as follows: Complete the table of actuarially fair insurance premiums that could be charged to each group separately if insurers knew which group the individual belonged to. Suppose now that the risk category is private information. What is the average riskiness of a person seeking insurance? What premium would insurers have to charge to break even? Will all individuals purchase insurance at this price? If not, what would be the composition of risks facing the insurer? Would the premium found in part b. be sufficient to cover the risks taken by the insurer? What will be the eventual price of insurance in the equilibrium and which groups will participate? Is this an efficient outcome?

Solution

The expected loss = 10, 000× risk

        If risk=0.2      Actuarially fair premium=10000*(0.2/10)=2000

        If risk=0.4      Actuarially fair premium=10000*(0.4/10)=4000

        If risk=0.6      Actuarially fair premium=10000*(0.6/10)=6000

        If risk=0.8      Actuarially fair premium=10000*(0.8/10)=8000

It is always lower than the willingness to pay since people exhibit risk aversion.

2. If all agents participate the chance of a loss in the population is 50%.

The insurance company will have to charge at least 5,000 to avoid a loss,

and in a competitive market this will be the price of insurance.

3. At this price the lowest risk 20% category would not participate as the premium is too high.

If only the 40%, 60%, 80% risks are in the market then the chance of a loss is 60%.

The insurer would need to charge 6,000.

4. Continuing as above, when the price is equal to 6,000 only the 60% and 80% risks remain with overall risk 70%.

The insurer would raise the premium to 7,000 driving out the 60% group and thus the equilibrium price will be 8,000 and only the highest risk agents will be insure.

5. This is not an efficient outcome because if insurance could be provided on actuarially fair terms to each group individually this would generate a surplus since they are risk averse.

Instead, the adverse selection problem leads to the collapse of the insurance market leaving many agents uninsured.