Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 30L / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

Use the optimizer for

Z_{0}

also with

Z_{1}

. Strict increase of

U

and a strict inequality on a positive-probability event give

u (Z_{1}) > u (Z_{0})

.

ii

Solution

If

X_{i}

optimizes

Z_{i}

, then

p X_{1} + (1 - p) X_{0}

is attainable because the wealth set is affine. Concavity gives

u (p Z_{1} + (1 - p) Z_{0}) \geq E U (p (X_{1} + Z_{1}) + (1 - p) (X_{0} + Z_{0})) \geq p u (Z_{1}) + (1 - p) u (Z_{0}) .

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