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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-43/5/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 43 / 5 / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

For

0 \leq s \leq 1

, pointwise concavity of the utility function gives

U ((s θ + (1 - s) η) X) \geq s U (θX) + (1 - s) U (η X) .

(1)

Taking expectations is legitimate because all three values are finite. Therefore $F (s θ + (1 - s) η) \geq s F (θ) + (1 - s) F (η)$ : $F$ is concave. Strict concavity is not required and is not asserted.

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