Notes on General Equilibrium

Posted on April 4, 2016
Tags:

Private Ownership Economies
Feasible Allocations and Pareto Optimality
Walrasian Equilibrium
Walrasian Equilibrium with Transfers
First Welfare Theorem
Second Welfare Theorem

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