A farmer believes that the probability that the next growing season will be "abnormally rainy" is p. With probability 1 - p, he believes that the next growing season will be "normally rainy", where 0 < p < 1. He can plant wheat or corn but not both. These crops have the following income prospects:
Crop YNR YAR
Wheat $28,000 $10,000
Corn $19,000 $15,000
Where, for a given crop, YNR and YAR represent the farmer's income when there is "normal rain" and "abnormal rain" respectively.
The farmer's utility function is U = ln(Y), where Y is income. Suppose the farmer must plant either wheat or corn but not both, and there is crop insurance available to farmers who only grow wheat. This crop insurance has a premium of $4,000 and pays the farmer $8,000 in an "abnormally rainy" season.
(i) Suppose p = 0.5. Determine the crop that the farmer will plant.
(ii) Is there a number, ̂p ∈ (0,1), such that if p > ̂p the farmer plants one crop and if p < ̂p, the farmer plants a different crop? If yes, determine ̂p. If no, explain why ̂p does not exist.