Question

Given: Suppose that 100 risk-averse individuals face a possible insurable health event of $10,000, and their degree of risk aversion is such that they are willing to pay a health insurance premium that is $1,250 higher than their expected healthcare costs. Assume 80 of the people are in low-risk subgroup and have a 25% probability of the event, and the other 20 people are in a high-risk subgroup and have a 75% probability of the event.

Assume individuals and the insurer know the size of the risk groups and the probability of a health event for each risk subgroup, but neither the individuals nor the insurer know which risk subgroup a particular individual is in.

If an insurer will not operate in the market unless its economic profit is greater than or equal to zero and it prices the health insurance premium according to the average for all 100 people (or the sub-group that will buy insurance at the equilibrium price), then the actuarially fair equilibrium premium plus $1,500 in loading charges per insured enrollee will be ____.

Answer 1

Expected Healthcare cost of the low risk subgroup = 0.25 * 10000 = 2500

So max payment propensity by an individual = $ 2500 + $ 1250 = $ 3750

Expected Healthcare cost of the high risk subgroup = 0.75 * 10000 = 7500

So max payment propensity by an individual = $ 7500 + $ 1250 = $ 8750

If the premium is x, then total revenue of the Insurer = 100 * x

So Expected payment by the insurer = 0.8 * 0.25 * 100 * 10000 + 0.2 * 0.75 * 100 * 10,000 = 350,000

So expected inflow of the insurer >=350,000

So Premium per person the insurer is expected to charge = 350,000/100 = 3,500

Adding loading costs, the total premium comes out to be = $ 3,5000 + $ 1500 = $5,000

Since the total premium is higher than the customer's willingness to pay, for low risk individuals, that subgroup will not take the insurance.

However, if low risk individuals do not take the insurance, then the expected payout of the insurer per insured individual becomes = 0.75 * 10,000 = 7500

To protect it's own interests, the insurer will want to charge above $7500 + 1500 = $ 9000.

This is again more than the reservation pricec of the high risk group. So this market will not hold and there will not be any insurer who would be willing to offer insurance.