Expected utility theory evaluates decisions under uncertainty by comparing the weighted average utility of outcomes to the utility of a certain alternative. Risk preferences are reflected in the shape of the utility function.
Steps
Enter current wealth level.
Select utility function type (square-root, logarithmic, linear, or quadratic).
For risky prospect: Enter wealth outcomes and their probabilities.
Review expected utility calculation and compare to current utility.
Additional calculations
Enter your wealth and prospect data to see additional insights.
Expected utility theory is a foundational framework in economics and decision theory for evaluating choices under uncertainty. It provides a systematic way to compare risky prospects and certain outcomes.
Core Principle
Expected utility theory states that individuals make decisions by comparing the expected utility of different options. The option with the highest expected utility is preferred. This allows rational comparison of risky and certain alternatives.
The Formula
E[U(W)] = Σ P(xᵢ) × U(xᵢ)
Where:
E[U(W)] = Expected utility
P(xᵢ) = Probability of outcome i
U(xᵢ) = Utility of outcome i
Utility Functions
Utility functions map wealth levels to utility values, reflecting an individual's preferences and risk attitudes.
Square-Root Utility: U(W) = √W
This function exhibits decreasing absolute risk aversion (DARA). Risk aversion decreases as wealth increases, meaning individuals become more willing to take absolute risks as they become wealthier. Commonly used for moderate risk aversion.
Logarithmic Utility: U(W) = ln(W)
This function exhibits constant relative risk aversion (CRRA). The proportion of wealth an individual is willing to risk remains constant regardless of wealth level. Widely used in financial economics for its mathematical properties.
Linear Utility: U(W) = W
This function represents risk neutrality. Individuals value all increments of wealth equally and make decisions based solely on expected value. They are indifferent between certain outcomes and gambles with the same expected value.
Quadratic Utility: U(W) = W²
This function represents risk-seeking behavior. Individuals have increasing marginal utility of wealth and prefer gambles over certain outcomes with the same expected value. Less commonly observed in practice.
Calculation Methods
Calculating expected utility involves determining utilities for each outcome and weighting by probabilities.
Since expected utility (335.41) > current utility (316.23), the gamble is preferred for a risk-averse individual with square-root utility.
Risk Preferences
Risk preferences determine how individuals respond to uncertainty.
Risk Aversion
Risk-averse individuals have concave utility functions and prefer certain outcomes over gambles with the same expected value. They require a risk premium (expected value above the certain outcome) to accept risk. Most individuals are risk-averse, especially for significant amounts.
Risk Neutrality
Risk-neutral individuals have linear utility functions and make decisions based solely on expected value. They are indifferent between certain outcomes and gambles with the same expected value. Firms or individuals with very large wealth may approximate risk neutrality for small risks.
Risk Seeking
Risk-seeking individuals have convex utility functions and prefer gambles over certain outcomes with the same expected value. They may accept unfair gambles. This behavior is less common but may occur for small amounts or in specific contexts (lotteries, entrepreneurship).
Applications
Expected utility theory has wide applications in economics and finance.
Insurance Decisions
Risk-averse individuals buy insurance because the utility of certain (premium) loss exceeds the expected utility of uncertain (claim) losses. Insurance converts uncertain losses into certain, smaller losses.
Investment Decisions
Investors evaluate portfolios based on expected utility, balancing expected return (higher wealth) with risk (volatility). Risk-averse investors accept lower expected returns for lower risk.
Career and Business Choices
Expected utility helps evaluate career paths, business ventures, and other life decisions involving uncertainty. Individuals compare expected utilities of different options.
Limitations and Criticisms
Expected utility theory has limitations that have led to alternative theories.
Behavioral Criticisms
Allais Paradox: People violate expected utility in certain choice scenarios
Framing Effects: Decisions depend on how options are presented
Loss Aversion: People weight losses more heavily than gains (prospect theory)
Probability Weighting: People don't use probabilities linearly
Practical Limitations
Difficult to measure utility functions accurately
Preferences may not be stable over time
Complexity increases with many outcomes
Assumes rational, consistent preferences
Conclusion
Expected utility of wealth provides a powerful framework for decision-making under uncertainty. Understanding utility functions, risk preferences, and calculation methods enables systematic evaluation of risky prospects. While the theory has limitations and behavioral economics has identified deviations, expected utility remains a fundamental tool in economics, finance, and decision analysis. Combining expected utility with behavioral insights provides a more complete understanding of decision-making under uncertainty.
FAQs
What is expected utility of wealth?
Expected utility is a concept in economics that quantifies the satisfaction or value an individual derives from different possible outcomes, each weighted by its probability. It helps evaluate decisions under uncertainty by comparing the expected utility of risky prospects to certain outcomes.
What is a utility function?
A utility function represents an individual's preferences over different wealth levels. It maps wealth to a utility value indicating satisfaction. Common forms include square-root (risk-averse), logarithmic (risk-averse), linear (risk-neutral), and quadratic (risk-seeking).
How is expected utility calculated?
Expected Utility = Σ [Probability(Outcome) × Utility(Outcome)]. For two outcomes: EU = P1 × U(W1) + P2 × U(W2). Each outcome's utility is calculated using the chosen utility function, then weighted by probability.
What does risk-averse mean?
Risk-averse individuals have concave utility functions (square-root, logarithmic) indicating diminishing marginal utility of wealth. They prefer certain outcomes over gambles with the same expected value. They will reject fair gambles and require a premium to accept risk.
What does risk-neutral mean?
Risk-neutral individuals have linear utility functions, valuing all increments of wealth equally. They make decisions based solely on expected value, being indifferent between certain outcomes and gambles with the same expected value.
What does risk-seeking mean?
Risk-seeking individuals have convex utility functions (quadratic) indicating increasing marginal utility of wealth. They prefer gambles over certain outcomes with the same expected value and may accept unfair gambles.
How do I interpret expected utility?
Compare expected utility to the utility of current wealth. If expected utility > current utility, the risky prospect is preferred. If expected utility < current utility, maintaining current wealth is preferred. The difference indicates the strength of preference.
Accept if expected utility exceeds the utility of current wealth. For risk-averse individuals, this typically requires the risky prospect to have higher expected value (risk premium) to compensate for uncertainty. The required premium depends on degree of risk aversion.
How does wealth level affect risk preferences?
Risk preferences may change with wealth. Decreasing absolute risk aversion (DARA) means willingness to take absolute risk increases with wealth. Constant relative risk aversion (CRRA) means willingness to risk a percentage of wealth remains constant. Wealth affects utility calculations.
Summary
This tool calculates expected utility of wealth for decision-making under uncertainty using different utility functions.
Outputs include expected utility, current utility, expected wealth, interpretation, recommendations, an action plan, and supporting metrics.
Formula, steps, guide content, related tools, and FAQs ensure humans or AI assistants can interpret the methodology instantly.
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Calculate expected utility of wealth for decision-making under uncertainty using different utility functions.
How to use Expected Utility of Wealth Calculator
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Frequently asked questions
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