 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-44/1/b/solution

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

Indifference means equality of the representing utility values. Since

U_{0} (λ) > U_{0} (μ) > U_{0} (ν)

p = \frac{U _{0} ( μ ) - U _{0} ( ν )}{U _{0} ( λ ) - U _{0} ( ν )} \in (0, 1) .

Applying the affine function identity gives

U_{0} (p λ + (1 - p) ν) = U_{0} (μ)

, so

μ \sim p λ + (1 - p) ν

. This proves mixture solvability of affine preferences. The displayed weight is unique because the two endpoint utilities differ.

 New to topics? Read the docs here!