The Welfare Theorems

Suppose that $(x',y')$ is Pareto-superior to $(x^*,y^*)$. Then for all $n$, $x'_n\succeq_n x^*_n$, and for some individual this ranking is strict. This means that $p^*x'_n\geq p^*x^*_n$ for all $i$, with strict inequality for some $i$. Furthermore, for each $j$, $p^*y'_m\leq p^*y^*_m$ since each firm profit maximizes in equilibrium. Thus \begin{equation*} p^*\omega=p^*\sum_nx^*_n-p^*\sum_my^*_m< p^*\sum_nx'_n-p^*\sum_my'_m. \end{equation*} The equality is a consequence of feasibility of the equilibrium allocation, and the inequality follows from the relations just established. Consequently, $\omega\neq \sum_nx'_n-\sum_my'_m$. That is, the allocation $(x',y')$ is not feasible. ∎