20.4: Expected Income, Expected Utility, and Risk Aversion II

John is evaluating a job offer from a company where his income will be uncertain. If the company performs well, John will earn an annual income of $81,000; otherwise, he will earn $49,000. It is assumed that either outcome has an equal chance, assigning a probability of 0.5 to each. This results in an expected income of $65,000. His decision-making is affected by the diminishing marginal utility of income.

John evaluates his options based on their utility. Expected utility accounts for risk preferences, unlike expected income. The expected utility is calculated as the sum of the utility values of possible incomes, weighted by their respective probabilities.

Expected utility = (P₁×U₁) + (P₂×U₂)

where P₁ and P₂ are the probabilities of the two outcomes, and U₁ and U₂ are their corresponding utility values.

For John,

• If the company performs well, his income will be $81,000 with a probability of 0.5, and a corresponding utility value of 9 units.
• If the company does not perform well, his income will be $49,000 with a probability of 0.5, and a corresponding utility value of 7 units.

Thus, John’s expected utility is (0.5×9) units +(0.5×7) units = 8 units

However, John can achieve the same utility of 8 units with a guaranteed income of $64,000. This reflects his preference for certainty over risk, even though the expected income in the uncertain scenario is higher.

The difference between the expected income of $65,000 and the guaranteed income of $64,000 represents John’s risk premium—$1,000. This is the amount he is willing to forgo to eliminate uncertainty and secure a guaranteed outcome. John’s preference for the guaranteed income highlights his risk-averse nature and demonstrates how diminishing marginal utility influences his decision.

Neil has a job offer with uncertain income. His expected income is $65,000. The diminishing marginal utility of income affects his preferences.

To understand this, the expected utility is calculated, which is the sum of the utility values of his possible incomes, weighted by their respective probabilities.

If Neil earns $81,000, his utility is 9 units. This is multiplied by its probability of 0.5.

If Neil earns $49,000, his utility is 7 units. This is multiplied by its probability of 0.5.

Adding these, Neil’s expected utility is 8 units.

However, Neil can achieve the same utility of 8 units with a guaranteed income of $64,000, as shown in the utility-income graph. 

Neil prefers a guaranteed $64,000 income over a higher but uncertain $65,000 expected income, showing his risk aversion.

The risk premium is the amount Neil is willing to sacrifice from his expected income, which in this case is $1,000, to eliminate the uncertainty in his income.

• Expected Income - The probable income considering probabilities of different outcomes.
• Utility - A measure of the satisfaction received from a particular choice.
• Expected Utility - The average utility of outcomes weighted by their probabilities.
• Risk Aversion - The preference to avoid risk and choose certainty.
• Risk Premium - The amount a person is willing to forgo for the certainty.

• Define Expected Income – Explain how to calculate it (e.g., expected income).
• Contrast Expected Income vs Utility – Explain the differences (e.g., risk averse).
• Explore Expected Utility – Describe its relevance to decision-making (e.g., risk premium).
• Explain Risk Preference – Briefly describe how risk aversion influences decisions.
• Apply Risk Premium – Understand its effect on a guaranteed income scenario.

• What is expected income and how to calculate it?
• What does expected utility indicate about a person's risk preference?
• How does a risk premium influence the choice between uncertain and guaranteed income?

• Students – Learn how risk aversion impacts income-related decision-making.
• Educators – Provides a clear understanding of expected income and utility in the decision-making process.
• Researchers – Relevance for research study in decision-making and risk assessment methods.
• Economic Enthusiasts – Offer insights into risk evaluation and preference.