# Notes on General Equilibrium

Posted on April 4, 2016
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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

1. $$y_j^{*}$$ 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}
2. $$x_i^{*}$$ 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}
3. 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 $$i$$.