# Notes on General Equilibrium

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### Table of contents

I took these notes while preparing for the placement exam for 14.123, the third quarter of the first-year “theory sequence” at MIT. It covers eseentially Part 4 of the this OCW course.

# Private Ownership Economies

A private ownership economy \(\mathcal{E}\) is a tuple \begin{equation} \label{eq:1} \mathcal{E} = (\{ (X_i, \succeq_i) \}_{i=1}^I, \{Y_j\}_{j=1}^J, \{(\omega_i, \theta_{i1}, \dots, \theta_{iJ})\}_{i=1}^I) \end{equation}where \(\theta_{ij} \in [0,1]\) and \(\sum_i \theta_{ij} = 1\) for each \(j = 1, \dots, J\). The interpretations are:

- \(X_i\) and \(\succeq_i\) are agent \(i\)’s consumption set and preferences over consumption.
- \(Y_j\) is the production set of firm \(j\).
- \(\omega_i\) is agent \(i\)’s endownment.
- \(\theta_{ij}\) is the shares of firm \(j\) held by agent \(i\).

It is assumed that \(X_i, Y_j \subseteq \mathbf{R}^L\), where \(L\) is the number of commodities. In addition, assume that \(\omega_i \in X_i\).

# Feasible Allocations and Pareto Optimality

An allocation is a vector \((x_1, \dots, x_I, y_1, \dots, y_J)\) such that \(x_i \in X_i\) and \(y_j \in Y_j\). A feasible allocation is an allocation \((x_1, \dots, x_I, y_1, \dots, y_J)\) such that \begin{equation} \label{eq:2} \sum_i x_i = \sum_i \omega_i + \sum_j y_j = \bar \omega + \sum_j y_j \end{equation}where \(\bar \omega = \sum_i \omega_i\).

A feasible allocation \((x_1, \dots, x_I, y_1, \dots, y_J)\) is Pareto Optimal if there is no other feasible allocation \((x'_1, \dots, x'_I, y'_1, \dots, y'_J)\) such that \begin{equation} \label{eq:3} x'_i \succeq x_i \quad \text{for every $i$ and strict for at least one $i$.} \end{equation}# Walrasian Equilibrium

An allocation \((x^{*}, y^{*}) \in \prod X_i \times \prod Y_j\) and a price vector \(p = (p_1, \dots, p_L)\) constitute a Walrasian equilibrium if

- \(y_j^{*}\) maximizes profits for firm \(j\) \begin{equation} \label{eq:4} p \cdot y_j \leq p \cdot y_j^{*} \quad \text{for every $y_j \in Y_j$} \end{equation}
- \(x_i^{*}\) maximizes utility for agent \(i\) within her budget set in that \begin{equation} \label{eq:5} x_i \succ x_i^{*} \implies x_i \notin B(p, \omega_i, y_j^{*}) := \big\{ x_i \in X_i : p \cdot x_i \leq p \cdot \omega_i + \sum_j \theta_{ij} p \cdot y_j^{*} \big\} \end{equation}
- The allocation is feasible \begin{equation} \label{eq:6} \sum x_i^{*} = \bar \omega + \sum_j y_j^{*}. \end{equation}

# Walrasian Equilibrium with Transfers

An allocation \((x^{*}, y^{*}) \in \prod X_i \times \prod Y_j\) and a price vector \(p = (p_1, \dots, p_L)\) constitute a Walrasian equilibrium with transfers if there is some \((w_1,\dots,w_I)\) such that the above conditions hold, except \eqref{eq:5} is replaced with \begin{equation} \label{eq:7} x_i^{*} \quad \text{is maximal in} \quad \{x_i \in X_i : p \cdot x_{i} \leq w_i\} \end{equation} A price quasi-equilibrium with transfers means that \eqref{eq:7} is replaced with \begin{equation} \label{eq:8} x_i \succ x_i^{*} \implies p \cdot x_i \geq w_i. \end{equation} In particular, any Walrasian equilibrium is a Walrasian equilibrium with transfers with \begin{equation} \label{eq:9} w_i = p \cdot \omega_i + \sum_j \theta_{ij} p \cdot y_j^{*} \end{equation}# First Welfare Theorem

If preferences are

- locally non-satiated

and if \((x^{*}, y^{*}, p)\) is a price equilibrium with transfers, then the allocation \((x^{*}, y^{*})\) is Pareto optimal.

# Second Welfare Theorem

If preferences are

- convex
- locally non-satiated

and production sets \(Y_j\) are

- convex

then for every Pareto optimal allocation \((x^{*}, y^{*})\), there is a price vector \(p\) such that \((x^{*}, y^{*}, p)\) is a price quasi-equilibrium with transfers. If there is a cheaper consumption bundle \(x'_i\) for agent \(i\) in that \(p \cdot x'_i < w_i\), then \eqref{eq:7} holds for agent \(i\).