Expected utility and certainty equivalents

What you will learn

Compare a sure final wealth with a finite lottery under linear and square-root utility, then reconcile the model’s certainty equivalent and risk premium.

Prerequisites: Expected value as a weighted average · Variance and standard deviation · Unequal outcome weights

A probability law is not a preference rule

A lottery specifies possible outcomes and their probabilities. A preference model says how those outcomes are valued. Maximizing expected final wealth is one rule, corresponding to linear utility u(w)=w. Expected-utility reasoning instead transforms each final wealth with a stated utility function and averages those utilities with the original probabilities. The probabilities do not change merely because the preference rule changes.

Use final wealth consistently. If a model already supplies final wealths 25 and 100, do not subtract an invented initial stake or reinterpret those numbers as incremental profits. Our activity uses positive educational wealth units and supplied linear or square-root preferences. It does not diagnose a learner’s risk tolerance, recommend a wager or assert that expected utility describes every real decision.

Transform before averaging

With equal probability on wealth 25 and 100, expected wealth is 62.5. For u(w)=√w, outcome utilities are 5 and 10, so expected utility is 7.5. Taking the square root of expected wealth instead would produce √62.5, a different number. The order of the transformation and expectation matters for nonlinear functions.

Compare a sure final wealth of 60 under the same utility rule. Linear utility favors the lottery because 62.5 exceeds 60. Square-root utility favors the sure amount because √60 exceeds 7.5. This reversal follows from the supplied preference functions; neither a subsequently lucky realization nor the larger mean settles which preferences someone ought to have.

Invert utility to return to wealth units

The certainty equivalent CE solves u(CE)=E[u(W)]. With square-root utility, CE=(E[√W])². In the equal-case example, CE=7.5²=56.25. This is a sure final wealth giving the same modeled utility as the lottery. It is not expected utility itself, not a realized result and not an amount that must be offered or paid.

For this lesson, the risk premium is E[W]−CE, giving 62.5−56.25=6.25. It measures the gap between mean wealth and the equivalent sure wealth in the stated model. Concavity explains why CE is no greater than the mean for this finite positive-wealth lottery. A different utility function, starting-wealth interpretation or probability law can change the result; a numerical premium is not an insurance quote or an empirical price.

Hands-on: Change preferences without changing the lottery

Allocate case probability, compare linear and square-root choices, and reconcile mean wealth, expected utility, certainty-equivalent wealth and premium.

Reading utility workshop…

Common mistakes

• Transforming the mean instead of averaging transformed outcomes.
• Comparing a utility value directly with a wealth amount.
• Treating the sure offer’s price as the risk premium.
• Inferring real preferences from a supplied utility function or one outcome.

Keep these ideas

• Specify both the outcome law and preference model.
• Transform each final wealth before averaging utility.
• Solve u(CE)=E[u(W)] to return to wealth units.
• State the baseline and utility assumptions behind any premium.

Further reading

• John Y. Campbell — Choice under Uncertainty (2018), sections 1.1–1.2 ↗

  Official Princeton University Press chapter excerpt: expected utility, concavity and certainty-equivalent definitions, printed pages5–8. Bounded section review, not the whole book. Our finite positive-wealth models and diagrams are original; no source exercises or illustrations reproduced. Checked 2026-10-21.