General equilibrium
§1. Exchange economy¶
Walrasian equilibrium¶
A Walrasian equilibrium for the economy \(\mathcal{E}\) is a vector \((p, (x^i)_{i\in \mathcal{I}})\) such that:
- 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}
- 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.