In Mongolia, winters can be of two types, "cold" or "very cold". If the winter is "very cold", there is a probability = 0.5 that a farmer loses 90% of her livestock (sheep, goats etc.). If the winter is just cold, all the farmer's livestock survive. The probability of "very cold" winters is p = 0.2 (and therefore the probability of "cold" winters is 0.8). In other words, each winter the farmer in Mongolia faces the following lottery

assuming that the insurance company cannot observe the action e chosen by the farmer. Now assume that the insurance company offers a different type of insurance (call it a catastrophic insurance) for the farmers in Mongolia. Before the beginning of the winter, the insurance company collects a premium P from the farmer. Then, if the winter is just "cold", the insurance company does not pay anything to the individual; if the winter is "very cold" the insurance company pays a coverage C to the individual, no matter if the farmer loses the (%90 of the) livestock or not (and no matter what is the hidden action e chosen by the farmer).

i. Calculate the fair premium for this insurance product. (Recall, the probability of "very cold" winter is p = 0.2)

ii. Do you think that this insurance policy presents a moral hazard problem? Explain.

iii. Illustrate the lotteries for the farmer if

She buys the insurance and chooses e = 0

She buys the insurance and chooses e = 1

(like before assume I=1000, L=900, =40, p=0.2, =0.5 and =0.25)

Now assume that a risk averse individual chooses to buy the fair insurance with coverage C = 900 and chooses also the action e = 1. Is the resulting allocation Pareto efficient? Why or why not? (keep assuming a risk averse individual and the risk neutral insurance company).