Welfare Theorem

Micro EC487_Advanced_Microeconomics

Before coming to the Welfare Theorem, it is important to recall the Pareto Optimality concept.

Proof of ⇒

$$(x^{\ast }{)}_{j=1}^N\text{ is Pareto Optimal }\exists p\in {\mathbb{R}}_{+}^L\text{ s.t. }{x}_j^{\ast }\text{ solves }\underset{x_j\in X_j}{\max }p\cdot x_j\forall j$$

Welfare Theorem

Long story short, there are two Welfare Theorem:

1. Competitive Equilibrium is Pareto Efficient.
2. Any Pareto Efficient allocation can be an equilibrium.

The first one: Suppose by contradiction, an improvement could be found

$$\exists \bar{x}\text{feasible}$$ $$\begin{array}{r}{u}^j({\bar{x}}^j)\ge u^j(x^{\ast j})⟹\sum _{\ell =1}{p}_{\ell }{\bar{x}}_{\ell }^j\ge \sum _{\ell =1}{p}_{\ell }{x}_{\ell }^{\ast j}\forall j\\ u^k({\bar{x}}^k)>u^k(x^{\ast k})⟹\sum _{\ell =1}{p}_{\ell }{\bar{x}}_{\ell }^k>\sum _{\ell =1}{p}_{\ell }{x}_{\ell }^{\ast k}\text{ for some }k\end{array}$$ $$\sum _{j=1}^N\sum _{\ell =1}^L{p}_{\ell }{e}_{\ell }^j\ge \sum _{j=1}^N\sum _{\ell =1}^L{p}_{\ell }{\bar{x}}_l^j\ge \sum _{j=1}^N\sum _{\ell =1}^L{p}_{\ell }{x}_{\ell }^{\ast j}=\sum _{j=1}^N\sum _{\ell =1}^L{p}_{\ell }{e}_{\ell }^j$$

If feasible, the total spent must be less than what was owned before.

To prove the second one, we need to apply Hyperplane Separation Theorem.

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• Hyperplane Separation Theorem