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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-40/5/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 40 / 5 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

A buyer accepts a posted price precisely when their valuation exceeds it; equality has probability zero. The independent uniform valuations therefore give probability

p^{2}

that neither accepts, for

0 \leq p \leq 1

. The seller's expected revenue is

R_{a} (p) = p (1 - p^{2}), R_{a}^{'} (p) = 1 - 3 p^{2} .

(1)

The interior critical point is

p = 1/ 3

, with negative second derivative; the endpoint revenues are zero. Prices above one make no sale and nonpositive prices cannot improve on this positive optimum. Hence

p_{*} = \frac{1}{3}, R_{a}^{*} = \frac{2}{3 3} .

(2)

Which willing buyer is served if both accept does not affect this revenue.

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