The Welfare Theorems

Suppose the $u_n$ are concave. The proof of the welfare theorems amounts to showing:

The First Welfare Theorem. If for all $i$, $x^*_n,\eta_n$ solves the first order conditions for $CE_n(x^*_n)$ with prices $p,$ then $x^*$ and multipliers $\lambda=\eta_1p$, $\nu_n=\eta_1/\eta_n$ solves the $PO(x^*)$ first order conditions.

Second Welfare Theorem. If $x^*$, $\nu$ and $\lambda$ solve $PO(x^*)$, then taking $\nu_1=1$, $x^*$ and multipliers $\eta_n=1/\nu_n$ and $p=\lambda$ solve all the $CE_n(x^*_n)$ first order conditions.

Due to concavity of the $u_n$, the first-order conditions are sufficient as well, and so for every interior Pareto optimal allocation there is a price that makes it a no-trade competitive equilibrium; and every competitive allocation is Pareto optimal.