Lange's Approach — Equilibrium
Now suppose an allocation $x'_1,\ldots,x'_I$ is a competitive equilibrium at price vector $p$.
Then
\begin{equation*}
\sum_nx'_n=\sum_n\omega_n,
\end{equation*}
supply equals demand, and for each $i$ the bundle $x'_n$ solves the optimization
problem
\begin{align*}
CE_n(\omega_n):\qquad\max{}&\> u_n(x_n)\\
\mbox{s.t.}\quad px_n&\leq p\omega_n.
\end{align*}
Again one can take the inequality to be an equality. The first order conditions include
\begin{equation*}
Du_n(x^*_n)=\eta_np
\end{equation*}
These conditions are necessary, and sufficient if the $U_n$ are concave.
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