General equilibrium

Walrasian model

A Walrasian equilibrium for the economy \(\mathcal{E}\) is a vector \((p, (x^i)_{i\in \mathcal{I}})\) such that:

1. Agents are maximizing their utilities: for all \(i \in \mathcal{I}\), \begin{equation} x^i \in \underset{c\in\mathcal{B}^i(p)}{\operatorname{argmax}} u^i(c) \end{equation}
2. Markets clear.

Welfare theorems

The First Welfare Theorem states that every Walrasian equilibrium allocation is Pareto optimal.

Let \((p, (x^i)_{i\in \mathcal{I}})\) be a Walrasian equilibrium for the economy. If agent's preferences are locally non-satiated, then the allocation \((x^i)_{i\in \mathcal{I}}\) is Pareto optimal.