general equilibrium theory - wordpress.com

General Equilibrium Theory

Jeffrey Ely

May 24, 2011

Jeffrey Ely General Equilibrium Theory

Motivation

Our study of markets focused on individual markets in isolation.

This was also true in 310-1.

Holding fixed the outcome in other markets, we analyzed the behaviorof participants in a single market.

But there are feedback loops across markets.I As the price of one good goes down, the demand for a substitute goes

up.I Opposite for complements.

To complete the study of competitive markets, we need a theory ofhow a market economy finds equilibrium in all markets simultaneously.


What’s Left

In the remaining lectures we will introduce the basic elements of thistheory.

I Pure exchangeI Competitive EquilibriumI The “Welfare Theorems”

These are the building blocks of many applications of economictheory.


Pure Exchange

To get a handle of the basic issues we will analyze a simple setting: pureexchange

There are many different goods traded in a market setting.

Different individuals enter the market holding differentcombinations/quantities of goods.

Everyone is a potential buyer and a potential seller of every good.

We are ignoring production.


Competitive Markets

Based on the ideas we have seen so far, we will develop a model with thefollowing elements

Private goods.

No externalities.

Price-taking behavior.

Market-clearing prices.


Model

N traders i = 1, . . . , N.

L goods, l = 1, . . . , L.

A separate market for each good. All markets operate simultaneously.

Endowments.I ei = (e1

i , e2i , . . . , eLi ) is i ’s endowment.

I e li ≥ 0 is a number indicating the quantity of good l that individual ienters the market with.

I Quantities are continuous variables.I e l = ∑N

i=1 e li is the total supply of good l in the economy.

Preferences.I After all trading is complete, traders leave the market with the bundles

of goods they have obtained.I They have preferences over bundles of goods represented by utility

functions ui .I If ωi = (ω1

i , . . . , ωLi ) is i ’s final bundle, then ui (ωi ) is i ’s utility.


Allocations

Definition

An allocation is a list of final bundles, one for each trader. We denote anallocation by ω where

ω = (ω1, . . . , ωN) is the list of bundles, and

ωi = (ω1i , . . . , ωL

i ) is the list of quantities in i ’s bundle.


Feasible Allocations

Definition

An allocation ω is feasible if it assigns no more than the total endowmentof each good, i.e. for every good l ,

N

∑i=1

ωli ≤

N

∑i=1

e li .


Pareto Efficiency

Definition

A feasible allocatin ω is Pareto efficient if there is no other feasibleallocation ω′ such that

ui (ω′i ) ≥ ui (ωi ) for all i and

uj (ω′j ) > uj (ωj ) for at least one j .


Prices and Demand

When trading is open, there will be market prices p = (p1, . . . , pL) foreach of the goods.

Each trader is assumed to act as a price-taker and make his plans forbuying and selling at these prices.

This translates to a familiar budget-constrained utility maximizationproblem.

The only twist is that “income” is itself determined by market prices.


“Income”

Suppose p = (p1, . . . , pL) is the list of market prices.

Trader i can sell his endowment.

He would earnL

∑l=1

ple li

He would then decide how to spend this money.


Spending Income

Definition

A bundle ωi = (ω1i , . . . , ωL

i ) is affordable for i at prices p if

L

∑l=1

plωli ≤

L

∑l=1

ple li


Maximizing Utility

Definition

A bundle ωi is a utility-maximizing demand at prices p if

ωi is affordable at prices p.

ui (ωi ) ≥ ui (ω′i ) for all bundles ω′i that are affordable at prices p.


Example With 2 Goods

The endowment of trader i .



For given prices p = (p1, p2), the “budget line” connects all bundles ωi

such thatp1ω1

i + p2ω2i = p1e1

i + p2e2i



Different prices would lead to different budget lines.


The Slope of the Budget Line

Since the equation for the budget line is:

p1ω1i + p2ω2

i = p1e1i + p2e2

i

We can call p1e1i + p2e2

i your “income” (at the prices p) and rewrite theequation as

ω2i = I /p2 −ω1

i (p1/p2).

The slope is −p1/p2. The ratio p1/p2 is called the relative price of p1.The relative price determines the rate at which good 1 can be “converted”into good 2.



So this budget line reflects a lower relative price for p1.



Than this one.



We can draw the indifference curve through the endowment point.



And we see that at these prices, the trader wishes to sell some of good 1and buy some more of good 2. The bundle he wishes to obtain is called hisdemand at prices p.



With a lower relative price of good 1



The trader will typically want to sell less of good 1.



And when the relative price is low enough, he may even switch to buyingmore of good 1 and selling good 2.


Market ClearingIf the prices are p and each trader i formulates his utility-maximizingdemand ωi , then for each good l ,

N

∑i=1

ωli

is the total demand for good l in the economy. If this total demandexceeds the total supply, i.e.

N

∑i=1

ωli >

N

∑i=1

e li

then there is excess demand for good l . Intuitively, the market price forgood l (relative to the other prices) is too low. And there is excess supplyof good l if

N

∑i=1

ωli <

N

∑i=1

e li

Intuitively, here the relative price of l is too low, and too many traders aretrying to sell good l .


Example with 2 Goods and 2 Traders: The Edgeworth Box

What you see here is called the Edgeworth Box.



We can illustrate the indifference curve of trader 1 through the endowmentpoint.



And also the indifference curve of trader 2.



The shaded area are all of those allocations that are feasible and bothtraders prefer to their endowments. These allocations Pareto dominate theendowment point.



Notice that these allocations involve trader 1 giving some quantity of good1 to trader 2 in exchange for some quantity of good 2.



Market prices determine a budget line. Since both traders face the samerelative prices, their budget line is the same.



This is a steep budget line, reflecting a high relative price of good 1.



When we find trader 1’s demand, we would expect to see him demand a lotof good 2, perhaps selling good 1.



But trader 2 may not demand as many units of good 1 as trader 1 wishesto sell. There is excess supply of good 1.



Similarly, there is excess demand for good 2.


Market Clearing

Definition

When the prices are p and the utility-maximizing demands at prices p areω = (ω1, . . . , ωN), then markets clear if

N

∑i=1

ωli =

N

∑i=1

e li

for all goods l = 1, . . . , L.


Price Adjustment to Clear Markets

Intuitively, the relative price of good 1 was too high. The supply of good 1at these prices was higher than demand and so. . .



competition among the sellers to find buyers would drive down the relativeprice of good 1.



If this does not yet clear the market, then the relative price of good 1 mustcontinue to fall until the market clears. This diagram illustrates marketclearing.


Competitive Equilibrium

The markets are in simultaneous equilibrium if prices are such that whentraders pursue their utility maximizing demands, all markets clear.

Definition

A competitive equilibrium is a list of prices p = (p1, . . . , pL) and anallocation ω = (ω1, . . . , ωN) satisfying

For each trader i , the bundle ωi is a utility-maximizing demand atprices p.

All markets clear.



This illustrates a competitive equilibrium.



Notice that the allocation that results Pareto dominates the endowment.



In fact the allocation that results is Pareto efficient.



Because these are the allocations that make 1 better off



And these are the allocations that make 2 better off. They have no pointin common.


Competitive Equilibrium Illustrated

Suppose trader 1 views the goods as perfect complements: u1(ω11, ω2

1) =min{ω1

1, 2ω22}. These indifference curves illustrate.



Suppose trader 2 views the goods as perfect substitutes: u2(ω12, ω2

2) =ω1

2 + ω22.



The Pareto efficient allocations are those on the vertex of 1’s indifferencecurves.



These prices are not a Competitive Equilibrium.



Because 2 demands ω2



And 1 demands ω1. There is excess demand for good 2. The problem isthat the price line is flatter than 2’s indifference curve, sending 2 to theedge of the box.



A competitive equilbirium price line must have the same slope as 2’s indif-ference curves.



Then this allocation is a utility maximizing demand for both traders. Andnotice that it is Pareto efficient and Pareto dominates the endowment point.This would be true no matter what the endowment point was.


Another Example

Suppose trader 1 also has linear indifference curves, but the slopes aredifferent.


Another Example

Now the Pareto efficient allocations are all of those on the top and left edgesof the box.


Another Example

Here is a price line which would lead to a Pareto efficient allocation whichPareto dominates the endowment point.


Another Example

The point ω represents a utility maximizing demand for 2, but not for 1.Trader 1 would like to sell more of good 1 and buy more of good 2, buttrader 2 has run out of good 2.


Another Example

The price line must coincide with 1’s indifference curve in order to make thetwo traders demand the same point.


More is Better

In all of the examples, a competitive equilibrium allocation is Paretoefficient. This is a general result. To prove it, we will add one assumptionto the model: more is better.

Assumption

If ωli ≥ ω̂l

i for all l with at least one strict inequality then ui (ωi ) > ui (ω̂i ).


Implication

Under the assumption that more is better prices cannot be negative and ifωi is a utility maximizing demand at prices p then

If ui (ω̂i ) ≥ ui (ωi ) then ∑l pl ω̂li ≥ ∑l ple li and

If ui (ω̂i ) > ui (ωi ) then ∑l pl ω̂li > ∑l ple li


The First Fundamental Theorem of Welfare Economics

Theorem

Under the assumption that more is better, any competitive equilbiriumallocation is Pareto efficient.


Proof

Suppose p is a competitive equilibrium price leading to allocation ω.Suppose that ω̂ Pareto dominates ω. We want to show that ω̂ is notfeasible.


Proof

If ω̂ Pareto dominates ω then

For all traders i ,ui (ω̂i ) ≥ ui (ωi )

And for at least one trader j ,

uj (ω̂j ) > uj (ωj )


Proof

From our implication of more is better:

For all traders i ,

∑l

pl ω̂li ≥∑

l

ple li

And for at least one trader j ,

∑l

pl ω̂lj > ∑

l

ple lj

We will add up these inequalities.


Proof

∑i

∑l

pl ω̂li > ∑

i∑l

ple li

∑l

∑i

pl ω̂li > ∑

l∑i

ple li

∑l

pl ∑i

ω̂li > ∑

l

pl ∑i

e li

∑l

pl (∑i

ω̂li −∑

i

e li ) > 0

Now pl ≥ 0 for all l , so there must be some good l for which

∑i

ω̂li > ∑

i

e li

which means that ω̂ is not feasible.


Discussion

Pareto Efficiency is a weak concept, so this is a weak statement.

For example, giving everything to one trader is Pareto efficient.

So we would like to know something more than this.

For example, what are the distributional possibilities from CE.


The Second Welfare Theorem

There is a second “fundamental theorem of welfare economics which saysthe following”

Theorem

Under some conditions, every Pareto efficient allocation can be achievedas a competitive equilibrium coupled with appropriate reallocation ofendowments.


Illustration of the Second Welfare Theorem

Suppose that 2 is “wealthier” than 1, in terms of their endowments. And wewould like to bring about a Pareto efficient allocation that is more equitable.



There is a price line that is tangent to these indifference curves.



If we required 2 to give some of his endowment to 1, moving to a pointon this line, then competitive equilibrium would move them to the targetallocation.


Summary

Competitive equilibrium leads to Pareto efficient allocations, and if equityis a concern then any Pareto efficient allocation can be achieved usingreallocations of endowments followed by trading in competitive markets.


We Are Done

It was fun. Good luck on the final. Have a Great Summer.


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