Risk Preferences & Utility

People differ in how they feel about risk. We model risk preferences using utility functions over wealth, $u (w)$ . The key insight: what matters is the expected utility of the gamble, not the utility of the expected value. A person is risk-averse if they prefer a certain amount to a gamble with the same expected value — formally, if $u (E (w)) > E (u (w))$ .

Key Term

Three Types of Risk Preferences

Risk-averse: $u (E (w)) > E (u (w))$ . Concave utility function (diminishing marginal utility of wealth). Prefers certainty. Most people are risk-averse.
Risk-neutral: $u (E (w)) = E (u (w))$ . Linear utility function. Indifferent between gamble and its $E V$ .
Risk-loving: $u (E (w)) < E (u (w))$ . Convex utility function. Prefers the gamble.

A concave utility function like $u = w$ exhibits diminishing marginal utility: each extra dollar is worth less

Graphically, a concave utility function curves below the chord connecting any two points. This means the utility of the expected value (a point on the curve) exceeds the expected utility (a point on the chord). Diminishing marginal utility of wealth drives risk aversion: losing 1,000 dollars hurts more than gaining 1,000 dollars helps, because you value money more when you have less of it.

Worked Example

A person has utility $u (w) = w$ . They face a gamble: 50% chance of $w = 100$ dollars, 50% chance of $w = 400$ dollars. Calculate $E (w)$ , $E (u (w))$ , and $u (E (w))$ . Is this person risk-averse?

Interactive — Concave Utility: Risk Aversion

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A concave utility-of-wealth curve with the gamble’s chord drawn across it.

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Can the same person be risk-averse for some gambles and risk-loving for others?

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Risk preferences and utility:

A risk-averse person has a utility function that is:

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