Questions: Expected Utility and Risk Aversion

Question 1

A person with utility u(w) = √w faces a gamble that pays $100 with probability 0.5 and $0 with probability 0.5. What is the certainty equivalent of the gamble?

• A. $50
• B. $25
• C. $10
• D. $5

Answer: (B)

Expected utility is 0.5 × √100 + 0.5 × √0 = 0.5 × 10 = 5. The certainty equivalent is the sure wealth with utility 5, so √CE = 5 and CE = $25. The expected value is $50, so the risk premium is $50 − $25 = $25.

Question 2

Which shape of utility-of-wealth function describes a risk-averse person?

• A. Linear
• B. Concave (marginal utility falls as wealth rises)
• C. Convex (marginal utility rises as wealth rises)
• D. Vertical

Answer: (B)

With a concave function, the utility of the expected value exceeds the expected utility of the gamble, u(E[w]) > E[u(w)]. The person therefore prefers a sure amount equal to the expected value. Linear utility gives risk neutrality and convex utility gives risk seeking.

Question 3

A risk-averse person will never pay more for insurance than the expected value of the loss it covers.

• T. True
• F. False

Answer: False

Because the person would rather have a certain outcome, they are willing to pay more than the expected loss to remove the risk. The most they will pay is the expected loss plus their risk premium. This is how insurers can cover costs and still sell policies.

Question 4

Explain why diminishing marginal utility of wealth implies risk aversion.

Model answer: With diminishing marginal utility, losing a given amount of money lowers utility more than gaining the same amount raises it. In a fair gamble the expected money gain is zero, but the expected utility change is negative, so the person prefers not to gamble and would accept a sure amount below the expected value instead.

Concavity of the utility function and diminishing marginal utility are the same property. Jensen's inequality, E[u(w)] < u(E[w]) for strictly concave u, states the result formally.