1. By Lemma;Conversely, suppose that equilibrium fails for some relevant supply
2. Since the aggregate-demand complex is n-dimensional, we may apply Cor. B.1: there exists a 0-cell C ? of L u {1,2} such that ? ? ? and which also satisfies x / ? D u {1,2} (p ? ) for p ? ? C ? . Moreover, C ? ? C ? (by Prop. 2.20(3)) and so C ? ? L u 1 ? L u 2 . Proof of Fact 5.6. Standard; see e;Cox;D u {1,2} (p ) is constant for any p ? C ? ? {1,2} , and thus both individual demand sets must be constant in C ? ? {1,2} . So C ? {1,2} ? L u 1 ? L u 2,2005
3. 11, emerging 0-cells get arbitrarily close to C for arbitrarily small . So we can choose sufficiently small that all intersection 0-cells for;Use the notation of Lemma B.12; using that lemma, choose ? > 0 sufficiently small so that, for all p ? B ? (p), both D u 1 (p ) = D u 1 * (p ) and D u 2 (p ) = D u 2 * (p )
4. But the 0-cells of L u {1 * ,2 * } are: those in C; and, if they exist, 0-cells at p of L u 1 * , L u 2 * . Take duals, and in the latter cases account for the demand of the other agent. ? 1 , ? 2 and ? {1,2} , respectively, for the demand complex cells at p of valuations u 1 , u 2 and u {1,2} . For any v and , if there exist no intersection 0-cells emerging from C then;) ? vol n (? {1,2} ),2002
5. Walrasian equilibrium in large, quasi-linear markets;M Azevedo;Theoretical Economics,2013