The First Welfare Theorem
The First Welfare Theorem gives conditions guaranteeing that a
competitive equilibrium allocation is Pareto optimal.
Recall that a preference order $\succeq_n$ is locally non-satiated at
$x^*_n$ if in every open neighborhood of $x^*_n$ there is an $x'_n\succ_nx^*_n$.
First Welfare Theorem. Let $\mathcal{E}$ be a private ownership economy
with an equilibrium $(p^*,x^*,y^*)$. Suppose for all $n$, $\succ_n$ is everywhere locally
non-satiated. Then $(x^*,y^*)$ is a Pareto-optimal allocation.
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