# Pure Exchange Model

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- Additional Contributors: None
Categories: CategoryModel CategoryMicro CategoryGeneralEquilibrium

- Related Models: None
Created: 2008-08-06

Suggested Citation: See license page.

Contents

## Introduction

This model assumes that there are no production possibilities in the economy, rather agents exchange their stock of initial endowments if there is the opportunity for mutually beneficial trade. The importance of wealth effects is shown since changes to the value of endowments affect the set of affordable alternatives. Analysis of this model can be used to prove the Second Fundamental Theorem.

## Model

There are two consumers ($$i=1,2$$) and two commodities ($$j=1,2$$) giving the consumption vector of consumer $$ i $$ as $$ x_i = (x_{i1},x_{i2}) $$ . Each consumer has an endowment vector $$ \omega_i = (\omega_{i1},\omega_{i2}) $$ where $$ \omega_{ij} \ge 0 $$ (assumed strictly positive) and the total endowment of each commodity in the economy is $$ \bar{\omega_j} = \omega_{1j} + \omega_{2j} $$ . An allocation is feasible if $$ x_{1j} + x_{2j} \le \bar{\omega_j} \forall j $$ .

A consumer's wealth is determined by the price vector $$ p = (p_1,p_2) $$ such that $$ p \cdot \omega_i = p_1\omega_{i1} + p_2\omega_{i2} $$ . This gives the budget set of consumer $$ i $$ as $$ B_i(p) = {p \cdot x_i \le p \cdot \omega_i} $$ and a budget line of slope $$ \frac{-p_1}{p_2} $$ .

Consumers have a preference revelation **symbol** over consumption vectors in this set, which are strictly convex, continuous and strongly monotone. For a given $$ p $$, the most preferred point in $$ B_i(p) $$ is expressed using the demand function $$ x_i(p,p \cdot \omega_i) $$ . As $$ p $$ varies, the offer curve is determined, which passes through $$ \omega_i $$ and every point on the offer curve is at least as preferred as the consumer's initial endowment.

## Solving

The Walrasian Equilibrium occurs when there is no excess demand in any markets. That is, there is a price vector $$ p^* $$ and an allocation $$ x^*=(x_1^*,x_2^*) $$ such that $$ x_i^* \succeq x_i'$$ $$ \forall x_i' \in B_i(p^*) $$ for $$ i=(1,2) $$ . Demand is homogenous of degree 0 in prices, therefore only the relative price $$ \frac{p_1^*}{p_2^*} $$ is determined in equilibrium.

An allocation $$ x $$ is Pareto optimal if there is no other feasible allocation $$ x' $$ where $$ x_i' \succeq x_i $$ for all $$ i $$ and $$ x_i' \succ x_i $$ for some $$ i $$ . Any Walrasian equilibrium necessarily belongs to the Pareto set, as with no excess demand one consumer cannot become better off without the other becoming worse off.

### Transfers

Since there may be multiple Pareto equilibria, it may be desirable for the social planner to select the 'socially optimal' allocation (using whichever decision mechanism has been chosen). This allocation can be achieved via competitive markets provided there is the appropriate inital endowment. The social planner therefore transfers resources between consumers (running a balanced budget such that $$ T_1+T_2=0 $$) to achieve this endowment level. This is the basis of the Second Fundamental Theorem, which states that any feasible Pareto equilibrium $$ x* $$ can be supported as an equilibrium with a price vector $$ p* $$ (as determined by competitive markets) and transfers $$ T $$ such that $$ x_i^* \succeq x_i' $$ for all feasible $$ x_i' $$ subject to $$ p* \cdot x_i' \le p* \cdot \omega_i + T_i $$ .

## Examples