 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-36/3/b/solution

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

(p^{*}, v^{*}) \in P \times Z

with

p^{*} \in g (v^{*})

and

v^{*} \in z (p^{*})

. The maximizing property of

g

and the assumed Walras law give

max_{i} v_{i}^{*} = max_{q \in P} q \cdot v^{*} = p^{*} \cdot v^{*} = 0.

Every coordinate of

v^{*}

is therefore nonpositive. Since

v^{*} \in z (p^{*})

, we obtain

z (p^{*}) \cap (- R_{+}^{L}) \neq = \emptyset .

This step does not yet imply that the entire vector is zero: a zero price can coexist with negative excess demand in its coordinate.

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