20.3: Expected Income, Expected Utility, and Risk Aversion I

Consider a hypothetical example where John is evaluating a job offer from a company. If the company performs well, John will earn an annual income of $81,000; if it performs poorly, he will earn $49,000. Each outcome is equally likely, with a probability of 0.5. These two outcomes are mutually exclusive, meaning only one can occur and their probabilities sum to 1. The amounts of $81,000 and $49,000 represent the payoffs associated with each outcome.

John's expected income is the average amount he expects to earn, considering all possible outcomes and their probabilities.

Expected Income = (P₁ × X₁) + (P₂ × X₂)

where P₁ and P₂ are the probabilities of the two outcomes, and X₁ and X₂ are their corresponding payoffs.

For John,

• If the company performs well, his income will be $81,000 with a probability of 0.5.
• If the company does not perform well, his income will be $49,000 with a probability of 0.5.

Thus, John’s expected income is (0.5 × $81,000) + (0.5 × $49,000) = $65,000

This analysis is helpful in decision-making under uncertainty. However, the expected utility should be calculated for further analysis because income alone does not capture how John values different income levels. To do this, the utility values corresponding to all possible income levels are required. It is assumed that John experiences diminishing marginal utility of income, meaning additional income leads to progressively smaller increases in satisfaction. The utility values for the different income levels are shown in the following table:

Table: Utility-Income Relationship

Consider a hypothetical scenario where Neil is offered a job at a company.

The income associated with the job is uncertain. If he performs well, he receives an annual salary of $81,000. Otherwise, his salary is $49,000. There is an equal probability of 0.5 for each outcome.

Expected income is calculated using the expected value analysis. 

Neil’s expected income is the product of 0.5 and $81,000 added to the product of 0.5 and  $49,000, resulting in $65,000.

To calculate the expected utility, the utility values corresponding to Neil's different income levels are required, which are shown on the graph.

Like most people, Neil experiences diminishing marginal utility of income. 

This analysis provides insights into how Neil's utility changes with income and sets the foundation for evaluating his expected utility under uncertainty.

• Expected Income - The average amount one expects to earn considering all possible outcomes and their probabilities.
• Income Outcome - The potential payoffs associated with different scenarios.
• Probability Factor - The likelihood of a specific outcome occurring.
• Diminishing Marginal Utility of Income- The concept that additional income leads to progressively smaller increases in satisfaction.
• Income Expected Meaning- What a person anticipates earning based on certain variables or factors.

• Define Expected Income - Explain what it represents and how it’s calculated (e.g., expected income formula).
• Contrast Income Outcome & Probability Factor – Explain the difference between potential payoffs vs their probability.
• Explore Examples – Describe scenario like, how John calculates his expected income (e.g., John's job offer).
• Explain Diminishing Marginal Utility of Income - Brief explanation on how additional income leads to smaller satisfaction increases.
• Apply in Context – Short explanation on how this analysis helps in decision-making under uncertainty.

• What is Expected Income and how is it calculated (Including the expected income formula)?
• How do you differentiate between Income Outcome and Probability Factor?
• What is Diminishing Marginal Utility of Income and how does it affect decision-making?

• Students – Understand how understanding 'Expected Income' supports financial learning.
• Educators – Provides a clear framework to teach concepts like Expected Income, Probability, and Utility.
• Researchers – Relevance for scientific research in behavioral economics.
• Finance Enthusiasts – Gain insights into decision-making processes under uncertainty.