Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 36 / 3 / b / Solution

By the Kakutani fixed-point theorem, choose $(p^{\ast },v^{\ast })\in P\times Z$ with $p^{\ast }\in g(v^{\ast })$ and $v^{\ast }\in z(p^{\ast })$. The maximizing property of $g$ and the assumed Walras law giveEvery coordinate of $v^{\ast }$ is therefore nonpositive. Since $v^{\ast }\in z(p^{\ast })$, we obtainThis step does not yet imply that the entire vector is zero: a zero price can coexist with negative excess demand in its coordinate.