Expected Income, Expected Utility, and Risk Aversion II

Expected utility is the sum of utility values for each possible income outcome, weighted by their respective probabilities. The formula is: Expected Utility = (P1×U1) + (P2×U2), where P represents probability and U represents utility. For example, if earning $81,000 yields 9 utility units with 0.5 probability and $49,000 yields 7 units with 0.5 probability, expected utility equals 8 units.

Diminishing marginal utility of income causes individuals to value certainty more than higher expected returns. A person can achieve the same utility from a guaranteed $64,000 as from an uncertain $65,000 expected income. This preference for certainty reflects risk aversion, where the psychological benefit of eliminating uncertainty outweighs the potential for additional income.

A risk premium is the amount an individual is willing to sacrifice from expected income to eliminate uncertainty. If someone has an expected income of $65,000 but would accept a guaranteed $64,000 instead, their risk premium is $1,000. This demonstrates how much value they place on certainty compared to the potential for higher but uncertain earnings.

Expected income is the mathematical average of possible outcomes weighted by probability, while expected utility accounts for individual risk preferences and diminishing marginal utility. Two scenarios with identical expected income can yield different expected utilities based on how the outcomes are distributed. Expected utility better reflects actual decision-making because it incorporates personal preferences for risk.

A utility-income graph shows that achieving the same utility level can require different income amounts depending on certainty. For a risk-averse individual, a guaranteed income of $64,000 provides equal utility to an uncertain income averaging $65,000. The curved shape of the utility function reflects diminishing marginal utility, where each additional dollar of income provides less satisfaction than the previous one.

Probabilities weight each outcome's utility value in the expected utility calculation. If two outcomes have equal probability (0.5 each), each contributes equally to the final expected utility. Unequal probabilities shift the weight toward more likely outcomes. For instance, a 0.7 probability outcome contributes more to expected utility than a 0.3 probability outcome, even if both have identical utility values.

Diminishing marginal utility means each additional dollar of income provides less satisfaction than the previous dollar. When evaluating uncertain job offers, this principle causes individuals to value the guaranteed portion of income more heavily. The difference between earning $81,000 and $49,000 creates unequal utility gains, making certainty more attractive than the higher expected income of $65,000.