Problem 8.4 examined a cost-sharing health insurance policy and showed that risk-averse individuals would prefer full coverage. Suppose, however, that people who buy cost-sharing policies take better care of their own health so that the loss suffered when they are ill is reduced from \(\$ 10,000\) to \(\$ 7,000 .\) Now what would be the actuarial fair price of a cost-sharing policy? Is it possible that some individuals might prefer the cost-sharing policy to complete coverage? What would determine whether an individual had such preferences? (A graphical approach to this problem should suffice.)

In a cost-sharing policy, the policyholder and the insurance company share the cost of a claim. In this case, we are given that people who buy cost-sharing policies take better care of their own health, reducing the potential loss from \(10,000 to \)7,000.

An actuarially fair price means that the premium charged by the insurance company is equal to the expected value of the loss. Let's denote the probability of getting ill as "p". To find the actuarial fair price for a cost-sharing policy, we can use the following formula: Actuarial Fair Price = p(Loss) In this case, the loss is $7,000, so the actuarial fair price for the cost-sharing policy would be: Actuarial Fair Price = p($7,000)

To determine whether some individuals might prefer the cost-sharing policy to complete coverage, we need to compare the actuarial fair prices for both policies and the reduced loss for cost-sharing policyholders. To achieve this, we can use a graphical approach by plotting the actuarial fair price and the expected loss on a graph. Remember that the probability of getting ill will determine the expected value of the loss for both policies, but for the cost-sharing policy, the reduced loss of $7,000 will also play a role in individual preferences.

On the graph, you will observe that some individuals might prefer the cost-sharing policy to complete coverage. Factors that determine such preferences include: 1. Cost-consciousness: Individuals who are more conscious about their insurance costs might prefer a cost-sharing policy due to its reduced loss and actuarial fair price. 2. Health awareness: People who take better care of their health or are less prone to illness might also prefer a cost-sharing policy as they will have lower expected loss. In conclusion, the actuarial fair price of a cost-sharing policy would be p times the reduced loss of $7,000. Depending on individual factors like cost-consciousness and health awareness, some people might prefer cost-sharing policies over full coverage.