Kat's preferences over levels of income are represented by the utility function illustrated in the figure above. When faced with a choice between a lottery, and a sure-thing outcome with the same expected value,

Question

BlackTom AI · Accepted Answer

A concave utility function implies risk aversion, so Kat prefers the sure thing.

1) Relevant concepts: utility as a function of income, risk aversion, Jensen’s inequality (for concave functions), and expected utility. A concave U(I) means marginal utility falls with income; for such U, the utility of the expected income is at least as large as the expected utility of a lottery over incomes.

2) Step-by-step reasoning:
- Let a lottery yield incomes I1 and I2 with probabilities p and 1−p. The expected income is Ī = p I1 + (1−p) I2.
- The expected utility of the lottery is EU = p U(I1) + (1−p) U(I2).
- The utility of receiving the sure amount Ī is U(Ī).
- Because the graph shows U(I) bending downward (concave), Jensen’s inequality gives U(Ī) > p U(I1) + (1−p) U(I2) for strict concavity.

3) Calculation example (intermediate steps):
- Pick I1 = 0, I2 = 100, p = 0.5 so Ī = 50.
- If U(I) = sqrt(I) (a concave function), U(50) = sqrt(50) ≈ 7.071.
- Expected utility of lottery = 0.5·U(0)+0.5·U(100) = 0.5·0 + 0.5·10 = 5.
- Since 7.071 > 5, the sure 50 yields higher utility than the lottery.

4) Why the correct answer is correct: The illustrated U(I) is increasing and concave, so Kat exhibits risk aversion; she prefers the certain outcome with the same expected value because it gives higher utility than the lottery’s expected utility.

5) Why other options are incorrect:
- “Kat prefers the lottery.” would require a convex utility (risk-seeking), which contradicts the concave curve.
- “Kat is indifferent” would require a linear utility (risk-neutral), not the curved shape shown.
- “Preferences cannot be determined…” is wrong because the curvature of the utility function precisely indicates risk attitude and determines choices between lotteries and sure amounts.

Conclusion: Kat prefers the sure thing, which matches the provided answer.

Kat's preferences over levels of income are represented by the utility function illustrated in the figure above. When faced with a choice between a lottery, and a sure-thing outcome with the same expected value,单项选择题

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