First Welfare Theorem

EC487_Advanced_Microeconomics

Theorem

Suppose that $u^j$ is continuous and increasing for all $j$. If the price vector $p^j\in {\mathbb{R}}_{+}^L$ is a Walrasian Equilibrium of the exchange economy $(N,e,u)$, then the corresponding allocation $(x^{\ast j}{)}_{j=1}^N$ where $x^{\ast j}=x^j(p^{\ast },w^j(p^{\ast }))$ is Pareto Optimality. (Pareto efficient).

Intuition

The intuition of this argument is true is that, suppose there is a Pareto Improvement over the equilibrium allocation. Then at least one consumer should strictly prefer the new bundle, which means it must strictly more costly than they could afford (otherwise the consumer would choose this new bundle in the first place). But then the total cost of the Pareto Improvement exceeds the total welfare, contracting the feasibility.

Proof

Todo

---

See Welfare Theorem