Danielle is a farmer, with a utility function of $\backslash ($ U$=$I$^\{0\mathrm{.}5\}\backslash ),$ where $\backslash ($ U $\backslash )$ is Danielle's utility, and $\backslash ($ I $\backslash )$ is her income. If the weather is good, she will earn $\backslash (\backslash$$ $100,000\backslash ).$ If there is a hailstorm, she will earn only $\backslash (\backslash$$ $50,000\backslash ).$ The probability of a hailstorm in any given year is $\backslash (30\backslash \%\backslash ).$

a$.$ What is Danielle's expected income if she is uninsured? Her expected utility? Round expected utility to one decimal place.

b$.$ Suppose a crop insurer makes the following offer to Danielle: In years when there is no hailstorm, Danielle pays the insurer $\backslash (\backslash$$ $16,000\backslash ).$ In years when there is a hailstorm, the insurer pays Danielle $\backslash (\backslash$$ $34,000\backslash ).$ What is Danielle's expected income? Her expected utility? Round the expected utility to one decimal place.

Expected income: $\backslash$$ Expected utility: c$.$ Consider the statement: "The insurance agreement in part b reduces Danielle's expected income. Therefore, it must make her worse off." This statement is because

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although the insurance agreement reduces Danielle's expected income, it raises her expected utility

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d$.$ Suppose instead the insurer offers Danielle the following: In years when there is no hailstorm, Danielle pays the insurer $\backslash (\backslash$$ $10,000\backslash )$; in years when there is a hailstorm, the insurer pays Danielle $\backslash (\backslash$$ $20,000\backslash ).$ What is Danielle's expected income? Her expected utility? Round the expected utility to one decimal place.

Expected income $\backslash$$:

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Expected utility:

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e$.$ Although Danielle's expected income in part $\backslash (\backslash$mathbf$\{$d$\}\backslash )$ is in part $\backslash (\backslash$mathbf$\{$b$\}\backslash ),$ her expected utility is

$.$ This reflects the fact that the agreement eliminates less income variation than the agreement in