91Ó°ÊÓ

Danielle is a farmer with a utility function of $$U=I^{0.5}$$ , 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)?

Step by step solution

01

Calculating Expected Income (Uninsured)

Danielle's income depends on weather conditions: - Earns \(100,000 with a 70% probability if the weather is good. - Earns \)50,000 with a 30% probability if there is a hailstorm.The expected income is calculated as:\[E(I) = 0.7 \times 100,000 + 0.3 \times 50,000 = 85,000.\]So, Danielle's expected income without insurance is $85,000.

02

Calculating Expected Utility (Uninsured)

Utility is derived using the income. Therefore, we calculate the utility for each income scenario:- Utility if weather is good: \( U_{100} = 100,000^{0.5} = 316.23 \).- Utility if a hailstorm: \( U_{50} = 50,000^{0.5} = 223.61 \).Expected utility is:\[E(U) = 0.7 \times 316.23 + 0.3 \times 223.61 = 287.62.\]Danielle's expected utility without insurance is 287.62.

03

Calculating Expected Income with Insurance in (b)

With the insurance plan in (b):- If no hailstorm: Earns \( 100,000 - 16,000 = 84,000 \).- If hailstorm: Earns \( 50,000 + 34,000 = 84,000 \).Expected income is:\[E(I) = 0.7 \times 84,000 + 0.3 \times 84,000 = 84,000.\]Danielle's expected income with this insurance is $84,000.

04

Calculating Expected Utility with Insurance in (b)

Utility for each scenario becomes:- Utility if no hailstorm: \( U_{84} = 84,000^{0.5} = 289.82 \).- Utility if hailstorm: \( U_{84} = 84,000^{0.5} = 289.82 \).Expected utility is:\[E(U) = 0.7 \times 289.82 + 0.3 \times 289.82 = 289.82.\]Danielle's expected utility with this insurance is 289.82.

05

Comparing the Insurance from (b) to No Insurance

The insurance policy in (b) reduces Danielle's expected income from $85,000 to $84,000, but it improves her expected utility from 287.62 to 289.82. The increase in utility shows that Danielle is better off due to risk reduction.

06

Calculating Expected Income with Insurance in (d)

For the insurance offer in (d):- If no hailstorm: \( 100,000 - 10,000 = 90,000 \).- If hailstorm: \( 50,000 + 20,000 = 70,000 \).Expected income is:\[E(I) = 0.7 \times 90,000 + 0.3 \times 70,000 = 84,000.\]Danielle's expected income is the same $84,000 as in part (b).

07

Calculating Expected Utility with Insurance in (d)

Utility for each scenario:- Utility if no hailstorm: \( U_{90} = 90,000^{0.5} = 300 \).- Utility if hailstorm: \( U_{70} = 70,000^{0.5} = 264.58 \).Expected utility is:\[E(U) = 0.7 \times 300 + 0.3 \times 264.58 = 288.37.\]Danielle's expected utility with this insurance is 288.37.

08

Comparing Outcomes

The insurance plan in (d) gives Danielle the same expected income as in (b), but higher expected utility than being uninsured (287.62) and less utility than insurance in (b) (289.82). Thus, (b) offers better risk reduction while (d) offers slightly better utility than being uninsured.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Risk Management

Understanding risk management is key to comprehending how individuals like Danielle make decisions about their financial future. Risk management involves identifying, assessing, and prioritizing risks, followed by the application of resources to minimize, control, and monitor the probability or impact of unfortunate events. In the context of Danielle, the risk is primarily related to the unpredictable weather conditions affecting her farming income.

In risk management, there are several strategies:

• Risk Avoidance: Eliminating exposure to certain risks, which can be impractical for farmers reliant on weather.
• Risk Reduction: Taking steps to reduce the severity of risks, such as using insurance.
• Risk Sharing: Distributing the risk among various parties, commonly achieved through insurance mechanisms.
• Risk Retention: Accepting the risk and its consequences, manageable through self-insurance or emergency funds.

In Danielle's case, her decision to consider insurance involves both risk sharing and risk reduction. She mitigates the financial impact of a hailstorm by sharing the risk with the insurer, thus securing her economic stability against natural uncertainties.

Insurance Economics

Insurance serves as an essential tool in economics, notably in providing financial protection against specific risks. In Danielle's situation, the insurance policy functions by equalizing her income irrespective of weather conditions, albeit at a cost.

Mechanics of Insurance:

• Premium Payments: These are regular payments made for coverage, usually less than the potential benefit.
• Coverage: Amount the insurer compensates in claims, which in this case matches her income whether or not a hailstorm occurs.
• Risk Pooling: Combining risks from many individuals, which allows insurers to predict and manage risk distributions effectively.

Insurance helps in stabilizing income when faced with risks. Though Danielle's expected income with insurance might be slightly lower, her financial predictability increases. This enhanced certainty typically outweighs the financial cost of reduced expected income, as seen from her improved utility with insurance.

Utility Functions

Utility functions are mathematical representations of preferences over a set of goods and outcomes. Danielle uses such a function to measure her satisfaction from different income levels. Her utility function, given as \(U = I^{0.5}\), reflects a diminishing marginal utility, which is typical in real-world preferences.

Key Features of Utility Functions:

• Diminishing Marginal Utility: Additional income provides less incremental satisfaction, making income smoothing via insurance appealing.
• Risk Aversion: This utility structure signifies that Danielle places a higher value on stable income rather than higher but variable earnings.
• Expected Utility: Calculated as the weighted sum of utilities from all possible outcomes, it helps in making informed decisions under uncertainty.

By comparing utility with and without insurance, Danielle evaluates the agreement's impact on her overall well-being. The increase in her expected utility when insured demonstrates the value she assigns to minimizing economic uncertainty.