A farmer believes that the probability that the next growing season will be "abnormally rainy" is . With probability 1 , he believes that the next growing season will be "normally rainy", where 0 < < 1. He can plant wheat or corn but not both. These crops have the following income prospects:

YAR

$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 = In(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 $4000 and pays the farmer $8000 of an "abnormally rainy" reason.

(i) Suppose = 0.5. Determine the crop that the farmer will plant.

(ii) Is there a number, (0,1), such that if > , the farmer plants one crop and if < , the

farmer plants a different crop? If yes, determine . If no, explain why does not exist.