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

For $v\in Z$, write $M(v)={\max }_i{v}_i$. Maximizing a linear functional on the price simplex givesThus $g(v)$ is a nonempty closed convex face of the compact price simplex. Its graph of a set-valued mapping is closed: if $v_n\to v$, $p_n\to p$ and $p_n\in g(v_n)$, then, for every fixed $q\in P$, the inequalities $p_n\cdot v_n\ge q\cdot v_n$ pass to $p\cdot v\ge q\cdot v$.

Each value $z(p)$ is closed in $Z$ by the closed-graph hypothesis, hence compact; it is nonempty and convex by assumption. Therefore $F(p,v)=g(v)\times z(p)$ has nonempty closed convex values in the nonempty compact convex set $K=P\times Z$.

Its graph of a set-valued mapping is also closed. If $(p_n,v_n)\to (p,v)$ and $(q_n,w_n)\to (q,w)$ with $q_n\in g(v_n)$ and $w_n\in z(p_n)$, the two closed-graph properties give $q\in g(v)$ and $w\in z(p)$. Finally, this closed graph implies upper hemicontinuity because the whole output space $K$ is compact. If an open set containing $F(p,v)$ failed to contain the nearby values, select outputs outside it along a convergent input sequence. A subsequence of those outputs converges in $K$; closedness puts its limit in $F(p,v)$, whereas closedness of the open set's complement puts it outside. This contradiction proves upper hemicontinuity.

Consequently all the hypotheses of the Kakutani fixed-point theorem hold: the domain is nonempty, compact and convex, and the set-valued mapping maps it into itself with nonempty closed convex values and upper hemicontinuity.

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.

Use the same Kakutani fixed-point theorem pair as in part (b). If any coordinate of $p^{\ast }$ vanished, the additional boundary condition would force some coordinate of every member of $z(p^{\ast })$, including $v^{\ast }$, to be positive. This contradicts $v^{\ast }\le 0$. Hence $p_i^{\ast }>0$ for every $i$.

Now Walras law reads ${\sum }_i{p}_i^{\ast }{v}_i^{\ast }=0$, a sum of nonpositive terms with strictly positive coefficients. Every term must vanish, soThe boundary hypothesis first excludes boundary prices; strict positivity of all prices then turns nonpositive excess demand into exact market clearing.