A household has wealth of $35,000 and may contract a serious disease. The possible loss from the disease is $10,000. An insurance company offers health insurance with a compensation of $K if the client contracts the disease, and charges an insurance premium of 0.1K. The cost of operation of the company is 0.01K^1.06, which exists regardless of whether the household contracts the disease or not.

a. (4 points) The insurance company estimates the chance of the client contracting the disease is 8%. What is its expected profit from selling the contract? What is the maximum amount of insurance (in terms of the amount of compensation) that the 2 company is willing to provide? (Hint: Consider the break-even point)

b. (5 points) Suppose the household only cares about her expected wealth (i.e., she is risk neutral), and she estimates that the chance of her getting the disease is 20%, how much insurance (in terms of the amount of compensation) would she purchase (demand) at the current price? How much insurance would she demand if her estimated chance of getting the disease is 10%?

c. (4 points) Suppose the household knows her condition better. Her estimated chance of 20% of getting the disease is more accurate. She chooses to withhold such information and purchase the insurance at her will. What would happen to the insurance company if it tries to maintain the price of 0.1K? (Hint: can the insurance company fulfill the demand of the household?)

d. (3 points) Following the information from the previous part, suppose the insurance company can change the price after viewing the demand from the household. What is the highest price it will charge that the client still accepts? Explain with supporting calculations.