Question

There are a continuum types of people in the society, whose health status can be represented by x, which lies in between 0 and 1. x follows uniform distribution U[0, 1]. A larger value of x indicates that the person’s health condition is better. The probability of getting sick and receiving medical treatment is 0.05 − 0.03x. The expenditure also depends on x, equal to 300 − 200x. For example, if x = 1, the probability of getting sick and receiving medical treatment is 0.05 − 0.03 = 0.02 and the expenditure equals to 300 − 200 = 100. If x = 0.5, the probability of getting sick and receiving medical treatment is 0.05 − 0.03 × 0.5 = 0.035 and the expenditure equals to 300 − 200 × 0.5 = 200. All people have an initial wealth of 500.

1. Now we consider one single person with x = 0.4. (15’)

(a) What is the maximum amount she is willing to pay for a health insurance policy that covers the medical treatment in full, if she is risk-neutral? (5’)

(b) How will your answer to question 1 change if she is risk averse with utility function u(w) = √ w? (5’)

(c) How will your answer to question (a) change if she is risk averse and the policy requires a coinsurance of 20% with a deductible of 50 (i.e., the buyer should pay 50 deductible plus 20% of the total medical expenditure in excess of the deductible)? (5’)

2. Now we consider a health insurance policy that covers the total expenditure in full. The premium level is 7. (20’)

(a) If all the people in the society are risk neutral. What is the fraction of people that will purchase this policy? Is the insurer’s profit positive, negative, or zero? [Hint: if X follows U[0, 1], the probability of X ≤ x0 when 0 ≤ x0 ≤ 1 is x0. ](10’)

(b) If all the people in the society are risk-averse, is the fraction of people that will purchase this policy larger or smaller than your answer to question 2(a)? How will the insurers’ profit change compared with your answer in question 2(a). (10’)

3. Now we consider a health insurance policy that requires a coinsurance of 20% with a deductible of 50. The premium level is 5. (15’)

(a) If all the people in the society are risk neutral. Denote x the fraction of people that will purchase this policy, please show an equation that x satisfies. Is the fraction of people that will purchase this policy x larger or smaller than your answer in 2(a)? (10’)

(b) Does raising the premium level always improve the insurer’s profit? Explain your answer. (5’)

Answer 1

In part (a), the maximum amount she is willing to pay for a health insurance policy that covers the medical treatment in full is calculated. In part (b), the maximum amount she is willing to pay as a risk-averse individual with a specific utility function is determined. In part (c), the impact of a coinsurance and deductible requirement on the maximum amount she is willing to pay is discussed. Part 2 focuses on the fraction of people that will purchase a health insurance policy under different risk scenarios and its impact on the insurer's profit. Part 3 explores the impact of raising the policy's premium level and its effect on the insurer's profit.

Step-by-step explanation:

(a) The maximum amount she is willing to pay for a health insurance policy that covers the medical treatment in full, if she is risk-neutral, can be calculated by finding the expected cost of getting sick and receiving medical treatment. The expected cost is equal to the probability of getting sick and receiving medical treatment (0.05 - 0.03x) multiplied by the expenditure (300 - 200x), and then subtracting it from her initial wealth of 500. So the maximum amount she is willing to pay can be calculated as: (0.05 - 0.03x) * (300 - 200x) - 500.

(b) If she is risk-averse with a utility function u(w) = √w, her willingness to pay for a health insurance policy will be influenced by her risk aversion. We can calculate her willingness to pay by finding the expected utility of her wealth after paying for the policy. The expected utility can be calculated as: E[u(w - premium)] = E[√(w - premium)]. To find the maximum amount she is willing to pay, we can solve for the premium that makes the expected utility equal to her initial utility level, which is √500.

(c) If she is risk-averse and the policy requires a coinsurance of 20% with a deductible of 50, her willingness to pay will be influenced by the coinsurance and deductible. The maximum amount she is willing to pay can be calculated by finding the expected cost after considering the coinsurance and deductible. The expected cost can be calculated as: (probability of getting sick and receiving medical treatment) * (expenditure above deductible * coinsurance rate) + deductible.

(a) To calculate the fraction of people that will purchase the health insurance policy if all the people in the society are risk-neutral, we need to find the probability of x ≤ x0, where 0 ≤ x0 ≤ 1. This probability is equal to x0. So the fraction of people that will purchase the policy is equal to the probability of x ≤ x0, which is x0.

(b) If all the people in the society are risk-averse, the fraction of people that will purchase the policy will be larger than the answer in 2(a). This is because risk-averse individuals are more likely to purchase insurance to protect themselves against uncertain health expenses. The insurer's profit will depend on the premium and the number of people purchasing the policy.

(a) If all the people in the society are risk-neutral, the equation that x satisfies can be obtained by setting the expected utility of wealth after purchasing the policy equal to the initial utility level. The fraction of people that will purchase this policy x is larger than the answer in 2(a) because the insurance policy includes a coinsurance and deductible.

(b) Raising the premium level does not always improve the insurer's profit. It depends on the elasticity of demand for the policy. If the increase in premium is greater than the decrease in the number of people purchasing the policy, the insurer's profit may decrease. The insurer needs to consider the price sensitivity of potential buyers when setting the premium level.