Problem: A farmer can grow wheat, or potatoes, or both. If the weather is good, an acre of land yields a profit of $2000 if devoted to wheat, and $1000 if devoted to potatoes. Should the weather be bad, an acre of wheat yields $1000 and potatoes $1750. Good and bad weather are equally likely.
a. Assuming that the farmer's utility of income, Y, is u(Y)= ln Y, what proportion of his land should he turn over to wheat?
b. Suppose the farmer can buy an insurance policy which pays $2 if the weather is bad and nothing if the weather is good, for each $1 of premium. How much insurance will he take out, and what proportion of his land will he devote to wheat? What would the answer be if the policy only paid $1.50 to compensate for the bad weather?