Lange's Approach — Optimality
The first order conditions are
\begin{align*}
Du_1(x^*_1)&=\lambda\\[6pt]
\nu_nDu_n(x^*_n)&=\lambda\text{ for $n\neq 1$}.
\end{align*}
for some $\lambda\in\R^L$ and scalars $\nu_n$, together with the
constraints. Strict monotonicity will imply that $\lambda,\nu\gg0$.
From this the $u_n$ the usual equality conditions for marginal
rates of substitution follow. These conditions, along with the
constraints, are necessary for an allocation to be Pareto
optimal. If we assume the $U_n$ are concave, they are sufficient
as well.
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