Lange's Approach — Optimality
If $x^*$ is an interior (strictly positive) Pareto optimal
allocation, then there is no reallocation that can increase the
utility of any consumer without decreasing the utility of anyone
else. let $u_n(x^*_n)=u^*_n$. Then $x^*$ solves the optimization
problem on $\prod_nX_n$:
\begin{align*}
PO(x^*):\qquad\max{}&\> u_1(x_1)\\[6pt]
\mbox{s.t.}\quad u_n(x_n)&\geq u^*_n\quad\mbox{for $i=2,\ldots,I$},\\[6pt]
\sum_nx_n&=\sum_nx^*_n.
\end{align*}
Assume that the $u_n$ are strictly increasing. Then the weak
inequalities can be assumed to be equalities. Let us, for
simplicity, consider an allocation in which each $x^*_n$ is interior
to $X_n$.
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