Danielle is a farmer, with a utility function of U = I0.5, where U is Danielle's utility and I is her income. If the weather is good, she will earn $100,000. If there is a hailstorm, she will earn only $50,000. The probability of a hailstorm in any given year is 30%.
a. What is Danielle's expected income if she is uninsured? Her expected utility?
b. Suppose a crop insurer makes the following offer to Danielle: In years when there is no hailstorm, Danielle pays the insurer $16,000. In years when there is a hailstorm, the insurer pays Danielle $34,000. What is Danielle's expected income? Her expected utility?
c. Comment on the following statement referring to your answers to parts (a) and (b): "The insurance agreement in (b) reduces Danielle's expected income. Therefore, it must make her worse off."
d. Suppose instead the insurer offers Danielle the following: In years when there is no hailstorm, Danielle pays the insurer $10,000; in years when there is a hailstorm, the insurer pays Danielle $20,000. How does Danielle's expected income and expected utility compare to the uninsured outcome in (a) and the insured outcome in (b)?