Problem
Consider an economy in which there are two types of risk-averse individuals. Type 1 risks losing $10 with 40 percent probability and nothing with 60 percent probability. Type 2 is in a riskier situation: with 80 percent probability, she loses $10, and with 20 percent probability she does not lose anything. Sixty percent of all individuals are of type 1, and 40 percent of type 2. Assume that the two types have the same utility function: u = (w0.6/0.6) where w is wealth. Both types of individuals are endowed with the same initial wealth w = $50. There is a risk-neutral insurer offering full insurance. This insurer is an NGO that just wants to break even, and suppose that there is no "loading factor" (i.e., no cost of providing insurance, so the insurer sets prices that are actuarially fair). The insurer can not distinguish between the two types, and thus has to charge the same premium to both types.
a. Compute the premium fee set by the insurer.
b. With this level of risk premium, which of the two types will purchase insurance? Explain your answer.
c. If the insurer anticipates that only individuals of type 2 will buy insurance, what is the premium charged in this case? Explain whether individuals of type 2 will ultimately buy insurance.