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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?

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Uncertainty, insurance

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?

Let x be the proportion of his land should he turn over to wheat, then the proportion of potatoes is (1-x).
When the weather is good, he can earn 2000x per acre from wheat, and 1000(1-x) per acre from potatoes. His total profit is then good Y=2000x +1000(1-x) = 1000+1000x.
However, when the weather is bad, he can earn 1000x per acre from wheat, and 1750(1-x) per acre from potatoes. His total profit is then bad Y=1000x +1750(1-x) = 1750 -750x.
The von Neumann-Morgenstern Utility function with uncertainty is