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 behavior
of participants in a single market.
But there are feedback loops across markets.
I
I
As the price of one good goes down, the demand for a substitute goes
up.
Opposite for complements.
To complete the study of competitive markets, we need a theory of
how a market economy finds equilibrium in all markets simultaneously.
Jeffrey Ely
General Equilibrium Theory
What’s Left
In the remaining lectures we will introduce the basic elements of this
theory.
I
I
I
Pure exchange
Competitive Equilibrium
The “Welfare Theorems”
These are the building blocks of many applications of economic
theory.
Jeffrey Ely
General Equilibrium Theory
Pure Exchange
To get a handle of the basic issues we will analyze a simple setting: pure
exchange
There are many different goods traded in a market setting.
Different individuals enter the market holding different
combinations/quantities of goods.
Everyone is a potential buyer and a potential seller of every good.
We are ignoring production.
Jeffrey Ely
General Equilibrium Theory
Competitive Markets
Based on the ideas we have seen so far, we will develop a model with the
following elements
Private goods.
No externalities.
Price-taking behavior.
Market-clearing prices.
Jeffrey Ely
General Equilibrium Theory
Model
N traders i = 1, . . . , N.
L goods, l = 1, . . . , L.
A separate market for each good. All markets operate simultaneously.
Endowments.
I e = (e 1 , e 2 , . . . , e L ) is i’s endowment.
i
i
i
i
I e l ≥ 0 is a number indicating the quantity of good l that individual i
i
I
I
enters the market with.
Quantities are continuous variables.
l
e l = ∑N
i =1 ei is the total supply of good l in the economy.
Preferences.
I
I
I
After all trading is complete, traders leave the market with the bundles
of goods they have obtained.
They have preferences over bundles of goods represented by utility
functions ui .
If ωi = (ωi1 , . . . , ωiL ) is i’s final bundle, then ui (ωi ) is i’s utility.
Jeffrey Ely
General Equilibrium Theory
Allocations
Definition
An allocation is a list of final bundles, one for each trader. We denote an
allocation by ω where
ω = (ω1 , . . . , ωN ) is the list of bundles, and
ωi = (ωi1 , . . . , ωiL ) is the list of quantities in i’s bundle.
Jeffrey Ely
General Equilibrium Theory
Feasible Allocations
Definition
An allocation ω is feasible if it assigns no more than the total endowment
of each good, i.e. for every good l,
N
N
i =1
i =1
Jeffrey Ely
General Equilibrium Theory
∑ ωil ≤ ∑ eil .
Pareto Efficiency
Definition
A feasible allocatin ω is Pareto efficient if there is no other feasible
allocation ω 0 such that
ui (ωi0 ) ≥ ui (ωi ) for all i and
uj (ωj0 ) > uj (ωj ) for at least one j.
Jeffrey Ely
General Equilibrium Theory
Prices and Demand
When trading is open, there will be market prices p = (p 1 , . . . , p L ) for
each of the goods.
Each trader is assumed to act as a price-taker and make his plans for
buying and selling at these prices.
This translates to a familiar budget-constrained utility maximization
problem.
The only twist is that “income” is itself determined by market prices.
Jeffrey Ely
General Equilibrium Theory
“Income”
Suppose p = (p 1 , . . . , p L ) is the list of market prices.
Trader i can sell his endowment.
He would earn
L
∑ pl eil
l =1
He would then decide how to spend this money.
Jeffrey Ely
General Equilibrium Theory
Spending Income
Definition
A bundle ωi = (ωi1 , . . . , ωiL ) is affordable for i at prices p if
L
∑ pl ωil ≤
l =1
Jeffrey Ely
L
∑ pl eil
l =1
General Equilibrium Theory
Maximizing Utility
Definition
A bundle ωi is a utility-maximizing demand at prices p if
ωi is affordable at prices p.
ui (ωi ) ≥ ui (ωi0 ) for all bundles ωi0 that are affordable at prices p.
Jeffrey Ely
General Equilibrium Theory
Example With 2 Goods
The endowment of trader i.
Jeffrey Ely
General Equilibrium Theory
Example With 2 Goods
For given prices p = (p 1 , p 2 ), the “budget line” connects all bundles ωi
such that
p 1 ωi1 + p 2 ωi2 = p 1 ei1 + p 2 ei2
Jeffrey Ely
General Equilibrium Theory
Example With 2 Goods
Different prices would lead to different budget lines.
Jeffrey Ely
General Equilibrium Theory
The Slope of the Budget Line
Since the equation for the budget line is:
p 1 ωi1 + p 2 ωi2 = p 1 ei1 + p 2 ei2
We can call p 1 ei1 + p 2 ei2 your “income” (at the prices p) and rewrite the
equation as
ωi2 = I /p 2 − ωi1 (p 1 /p 2 ).
The slope is −p 1 /p 2 . The ratio p 1 /p 2 is called the relative price of p 1 .
The relative price determines the rate at which good 1 can be “converted”
into good 2.
Jeffrey Ely
General Equilibrium Theory
Example With 2 Goods
So this budget line reflects a lower relative price for p 1 .
Jeffrey Ely
General Equilibrium Theory
Example With 2 Goods
Than this one.
Jeffrey Ely
General Equilibrium Theory
Example With 2 Goods
We can draw the indifference curve through the endowment point.
Jeffrey Ely
General Equilibrium Theory
Example With 2 Goods
And we see that at these prices, the trader wishes to sell some of good 1
and buy some more of good 2. The bundle he wishes to obtain is called his
demand at prices p.
Jeffrey Ely
General Equilibrium Theory
Example With 2 Goods
With a lower relative price of good 1
Jeffrey Ely
General Equilibrium Theory
Example With 2 Goods
The trader will typically want to sell less of good 1.
Jeffrey Ely
General Equilibrium Theory
Example With 2 Goods
And when the relative price is low enough, he may even switch to buying
more of good 1 and selling good 2.
Jeffrey Ely
General Equilibrium Theory
Market Clearing
If the prices are p and each trader i formulates his utility-maximizing
demand ωi , then for each good l,
N
∑ ωil
i =1
is the total demand for good l in the economy. If this total demand
exceeds the total supply, i.e.
N
∑ ωil >
i =1
N
∑ eil
i =1
then there is excess demand for good l. Intuitively, the market price for
good l (relative to the other prices) is too low. And there is excess supply
of good l if
N
∑ ωil <
i =1
N
∑ eil
i =1
Intuitively, here the relative price of l is too low, and too many traders are
trying to sell good l.
Jeffrey Ely
General Equilibrium Theory
Example with 2 Goods and 2 Traders: The Edgeworth Box
What you see here is called the Edgeworth Box.
Jeffrey Ely
General Equilibrium Theory
Example with 2 Goods and 2 Traders: The Edgeworth Box
We can illustrate the indifference curve of trader 1 through the endowment
point.
Jeffrey Ely
General Equilibrium Theory
Example with 2 Goods and 2 Traders: The Edgeworth Box
And also the indifference curve of trader 2.
Jeffrey Ely
General Equilibrium Theory
Example with 2 Goods and 2 Traders: The Edgeworth Box
The shaded area are all of those allocations that are feasible and both
traders prefer to their endowments. These allocations Pareto dominate the
endowment point.
Jeffrey Ely
General Equilibrium Theory
Example with 2 Goods and 2 Traders: The Edgeworth Box
Notice that these allocations involve trader 1 giving some quantity of good
1 to trader 2 in exchange for some quantity of good 2.
Jeffrey Ely
General Equilibrium Theory
Example with 2 Goods and 2 Traders: The Edgeworth Box
Market prices determine a budget line. Since both traders face the same
relative prices, their budget line is the same.
Jeffrey Ely
General Equilibrium Theory
Example with 2 Goods and 2 Traders: The Edgeworth Box
This is a steep budget line, reflecting a high relative price of good 1.
Jeffrey Ely
General Equilibrium Theory
Example with 2 Goods and 2 Traders: The Edgeworth Box
When we find trader 1’s demand, we would expect to see him demand a lot
of good 2, perhaps selling good 1.
Jeffrey Ely
General Equilibrium Theory
Example with 2 Goods and 2 Traders: The Edgeworth Box
But trader 2 may not demand as many units of good 1 as trader 1 wishes
to sell. There is excess supply of good 1.
Jeffrey Ely
General Equilibrium Theory
Example with 2 Goods and 2 Traders: The Edgeworth Box
Similarly, there is excess demand for good 2.
Jeffrey Ely
General Equilibrium Theory
Market Clearing
Definition
When the prices are p and the utility-maximizing demands at prices p are
ω = (ω1 , . . . , ωN ), then markets clear if
N
N
i =1
i =1
∑ ωil = ∑ eil
for all goods l = 1, . . . , L.
Jeffrey Ely
General Equilibrium Theory
Price Adjustment to Clear Markets
Intuitively, the relative price of good 1 was too high. The supply of good 1
at these prices was higher than demand and so. . .
Jeffrey Ely
General Equilibrium Theory
Price Adjustment to Clear Markets
competition among the sellers to find buyers would drive down the relative
price of good 1.
Jeffrey Ely
General Equilibrium Theory
Price Adjustment to Clear Markets
If this does not yet clear the market, then the relative price of good 1 must
continue to fall until the market clears. This diagram illustrates market
clearing.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium
The markets are in simultaneous equilibrium if prices are such that when
traders pursue their utility maximizing demands, all markets clear.
Definition
A competitive equilibrium is a list of prices p = (p 1 , . . . , p L ) and an
allocation ω = (ω1 , . . . , ωN ) satisfying
For each trader i, the bundle ωi is a utility-maximizing demand at
prices p.
All markets clear.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium
This illustrates a competitive equilibrium.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium
Notice that the allocation that results Pareto dominates the endowment.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium
In fact the allocation that results is Pareto efficient.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium
Because these are the allocations that make 1 better off
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium
And these are the allocations that make 2 better off. They have no point
in common.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium Illustrated
Suppose trader 1 views the goods as perfect complements: u1 (ω11 , ω12 ) =
min{ω11 , 2ω22 }. These indifference curves illustrate.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium Illustrated
Suppose trader 2 views the goods as perfect substitutes: u2 (ω21 , ω22 ) =
ω21 + ω22 .
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium Illustrated
The Pareto efficient allocations are those on the vertex of 1’s indifference
curves.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium Illustrated
These prices are not a Competitive Equilibrium.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium Illustrated
Because 2 demands ω2
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium Illustrated
And 1 demands ω1 . There is excess demand for good 2. The problem is
that the price line is flatter than 2’s indifference curve, sending 2 to the
edge of the box.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium Illustrated
A competitive equilbirium price line must have the same slope as 2’s indifference curves.
Jeffrey Ely
General Equilibrium Theory
Competitive Equilibrium Illustrated
Then this allocation is a utility maximizing demand for both traders. And
notice that it is Pareto efficient and Pareto dominates the endowment point.
This would be true no matter what the endowment point was.
Jeffrey Ely
General Equilibrium Theory
Another Example
Suppose trader 1 also has linear indifference curves, but the slopes are
different.
Jeffrey Ely
General Equilibrium Theory
Another Example
Now the Pareto efficient allocations are all of those on the top and left edges
of the box.
Jeffrey Ely
General Equilibrium Theory
Another Example
Here is a price line which would lead to a Pareto efficient allocation which
Pareto dominates the endowment point.
Jeffrey Ely
General Equilibrium Theory
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, but
trader 2 has run out of good 2.
Jeffrey Ely
General Equilibrium Theory
Another Example
The price line must coincide with 1’s indifference curve in order to make the
two traders demand the same point.
Jeffrey Ely
General Equilibrium Theory
More is Better
In all of the examples, a competitive equilibrium allocation is Pareto
efficient. This is a general result. To prove it, we will add one assumption
to the model: more is better.
Assumption
If ωil ≥ ω̂il for all l with at least one strict inequality then ui (ωi ) > ui (ω̂i ).
Jeffrey Ely
General Equilibrium Theory
Implication
Under the assumption that more is better prices cannot be negative and if
ωi is a utility maximizing demand at prices p then
If ui (ω̂i ) ≥ ui (ωi ) then ∑l p l ω̂il ≥ ∑l p l eil and
If ui (ω̂i ) > ui (ωi ) then ∑l p l ω̂il > ∑l p l eil
Jeffrey Ely
General Equilibrium Theory
The First Fundamental Theorem of Welfare Economics
Theorem
Under the assumption that more is better, any competitive equilbirium
allocation is Pareto efficient.
Jeffrey Ely
General Equilibrium Theory
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 not
feasible.
Jeffrey Ely
General Equilibrium Theory
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 )
Jeffrey Ely
General Equilibrium Theory
Proof
From our implication of more is better:
For all traders i,
∑ pl ω̂il ≥ ∑ pl eil
l
l
And for at least one trader j,
∑ pl ω̂jl > ∑ pl ejl
l
l
We will add up these inequalities.
Jeffrey Ely
General Equilibrium Theory
Proof
∑ ∑ pl ω̂il > ∑ ∑ pl eil
i
i
l
∑ ∑ pl ω̂il >
l
i
l
∑p ∑
ω̂il
−∑
eil )
i
l
∑ p (∑
l
l
ω̂il
i
l
∑ ∑ pl eil
l
i
l
> ∑p
l
∑ eil
i
>0
i
Now p l ≥ 0 for all l, so there must be some good l for which
∑ ω̂il > ∑ eil
i
i
which means that ω̂ is not feasible.
Jeffrey Ely
General Equilibrium Theory
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.
Jeffrey Ely
General Equilibrium Theory
The Second Welfare Theorem
There is a second “fundamental theorem of welfare economics which says
the following”
Theorem
Under some conditions, every Pareto efficient allocation can be achieved
as a competitive equilibrium coupled with appropriate reallocation of
endowments.
Jeffrey Ely
General Equilibrium Theory
Illustration of the Second Welfare Theorem
Suppose that 2 is “wealthier” than 1, in terms of their endowments. And we
would like to bring about a Pareto efficient allocation that is more equitable.
Jeffrey Ely
General Equilibrium Theory
Illustration of the Second Welfare Theorem
There is a price line that is tangent to these indifference curves.
Jeffrey Ely
General Equilibrium Theory
Illustration of the Second Welfare Theorem
If we required 2 to give some of his endowment to 1, moving to a point
on this line, then competitive equilibrium would move them to the target
allocation.
Jeffrey Ely
General Equilibrium Theory
Summary
Competitive equilibrium leads to Pareto efficient allocations, and if equity
is a concern then any Pareto efficient allocation can be achieved using
reallocations of endowments followed by trading in competitive markets.
Jeffrey Ely
General Equilibrium Theory
We Are Done
It was fun. Good luck on the final. Have a Great Summer.
Jeffrey Ely
General Equilibrium Theory