Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 44 / 1 / b

Table of contents
- Solution b

Solution

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.

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