Assume that the probability of the event “Little rain” is equal to α = 1 10 (and the probability of the event “Normal rain” is 1 − α = 9 10 ). If the farmer is risk neutral, will the farmer invest or not? Explain
(e) Stop assuming that the farmer is risk neutral, but keep the same assumptions as in (1d). Assume that the farmer has preferences represented by the utility over monetary outcomes u(x) = 120 − 8000 x (1) Will a farmer with the above preferences invest or not? Explain
(f) Now assume that a (risk neutral) insurance company offers the policy that pays $C if the event “Little rain” occurs. Assume that the policy is fairly priced: How much is the premium that the insurance company charges?
(g) Keep assuming that insurance company offers a fairly priced policy. However the insurance company does not fix the coverage C, but lets the customer choose C. Under these assumptions, will the farmer with preferences described in the point (1e) invest or not? Will he buy insurance? If so, how much coverage C will he choose?
(h) Now assume that there are two types of farmers: The farmer we met before (call him L) and a farmer who lives in a slightly different microclimate (call this farmer H). Assume that farmer H is identical to farmer L apart for one thing: In the microclimate in which farmer H lives, the probability of the event “Little rain” is equal to αH = 0.2. If the insurance company can distinguish between the two farmers and offer different insurance policies, what is the fair premium for the farmer H? Will he buy insurance? How much? will he invest?
(i) Now assume that the insurance company cannot distinguish between the two types of farmers. And assume that the proportion of H farmers is equal to 0 < q < 1. Without making any calculations can you explain why in such a situation we can have “adverse selection”. Be as clear as you can.