The Welfare Theorems

The Second Welfare Theorem. Let $(x^*,y^*)$ be a Pareto Optimal allocation for a private ownership economy $\mathcal{E}$ with the properties that

1. for all $n$, $X_n$ is convex,
2. the sets $R(x^*_n)$ are convex,
3. for some consumer $k$, $P(x^*_k)$ is convex and $\succeq_k$ is locally non-satiated at $x^*_k$,
4. $Y$ is convex.

Then there is a $p^*$ such that $(x^*,y^*,p^*)$ is a quasi-equilibrium for $\mathcal{E}$.