Lecture 2:
Competitive Equilibrium
Readings: Leach, Chapters 2 and 3
Competitive Equilibrium
Q: What kinds of social arrangements cause private
(self) interests to become aligned with the public
(collective) interest?
A: Adam Smith’s central thesis in the Wealth of
Nations (1776) was that if private property rights
are protected by the state, a competitive market
provides a social arrangement that causes the
pursuit of private interests (profit) to promote the
public interest (national prosperity).
Competitive Equilibrium
Q: How do economists prove Smith’s theory today?
A: Competitive General Equilibrium and the First Welfare Theorem.
A 2x2 Model:
• Two commodities ale (a) and bread (b)
• Two participants George (G) and Harriet (H)
• An endowment is a list of goods initially possessed by each person
• Ale Endowment:
• Bread Endowment:
• A feasible allocation is a list of goods that could finally possessed
by each person if the endowment is redistributed.
Allocation = (aG , bG , aH , bH) such that a  aG  a H and b  bG  bH
Competitive Equilibrium
•
Given some endowment, the set of all feasible allocations is the Edgeworth Box.
The endowment (E) and any allocation (A) are represented as points in the
Edgeworth Box. The endowment is sometimes referred to as the initial allocation.
aH
bG
A
aG
OH
bH
E
bG
OG
aH
bH
aG
Q: Why might society seek to choose a different allocation than the endowment?
A: The endowment may be viewed as unfair and most certainly is inefficient.
Competitive Equilibrium
Q: Why is the endowment inefficient?
A: Consider some arbitrary endowment (E). If the indifference curves through this point
form a lens, then there exists a feasible reallocation within the lens that makes both
people better off. The endowment E is inefficient in the sense that it wastes an
opportunity to make people better off without anyone having to pay the cost of
becoming worse off. Notice that inefficiency is linked to satisfaction and not to output.
Allocations within the lens make at least one person better off, without making
anyone worse off. Allocations outside the lens make at least one person worse off.
aH
OH
E
bG
U1
V1
OG
aG
bH
Competitive Equilibrium
Q: What is an efficient allocation?
A: If an inefficient allocation has a lens of available reallocations that make everyone
better off, then an efficient allocation is one in which there is no such lens of
opportunity. This occurs when the indifference curves touch at a point of tangency. At
such a point, the allocation is said to be Pareto optimal (or Pareto efficient).
aH
OH
P
E
bG
U1
V1
OG
aG
bH
Competitive Equilibrium
Q: Can this idea be generalized for an economy with many consumers, many
goods, and production?
Definition: An allocation is Pareto optimal if there is no alternative feasible
allocation in which one person is better off and no one worse off.
Another way of thinking about Pareto optimality is to introduce the idea of a
Pareto improvement.
Definition: A Pareto improvement is a feasible reallocation which makes at
least one person better off without making anyone worse off.
If a Pareto improvement exists for some allocation, then the allocation is not
Pareto optimal (inefficient). If no Pareto improvements exist, then the allocation is
Pareto optimal (efficient).
Competitive Equilibrium
Q: How many Pareto optimal allocations are there in our 2x2 Exchange Economy?
A: An infinite number of points in the Edgeworth Box have the property of
tangent indifference curves. This can be represented by the Contract Curve.
aH
OH
bH
P
E
bG
OG
aG
Competitive Equilibrium
Q: Suppose a competitive market spontaneously emerged in our 2x2
economy. What would be the impact of this market?
A: The market would produce a price for bread (Pb) and a price for ale (Pa).
Each individual (G and H) would respond by either selling bread to buy ale
or selling ale to buy bread. This decision would depend on their
preferences, their endowment, and the relative price of ale.
Q: What is the relative price of ale?
A: It is the price of a pint of ale in terms of loafs of bread sacrificed to
purchase the ale. If the price of bread is $2 per loaf and the price of ale is
$4 per pint, then the relative price of ale is 2 loafs of bread. This is the
bread price of ale. The relative price is a function of the money prices:
Relative Price of Ale = P = Pa / Pb = ($/pint of ale) / ($/loaf of bread)
= loafs of bread / pint of ale
Competitive Equilibrium
Q: Given a set of relative prices and the individual’s
endowment what will they do?
A: In terms of the Edgeworth box, the relative price
provides the slope of the budget constraint that goes
through the initial endowment. The budget constraint
presents each person’s set of feasible market exchanges
given their endowment and market prices. The person
will choose the point on the budget constraint that
maximizes their utility. This choice is the unique point on
the budget line where:
MRS = Pa / Pb = P = bread price of ale = slope of the budget line
Competitive Equilibrium
G wishes to sell aG  aG units of ale and buy bG  bG units of bread given the market
price. H wishes to sell bH  bH units of bread and buy aH  aH units of ale given the
market price P. The result is that P creates an excess supply of bread and an excess
demand for ale: Excess Supply of b  bH  bG and Excess Demand of a  a H  aG .
aH
aH
U1
OH
U2
P
1
bH
bG
bG
E
V1
V2
OG
aG
aG
bH
Competitive Equilibrium
Q: If there is an excess supply of bread (b) and an excess demand for
ale (a) how will the market respond?
A: The money price of ale will rise and the money price of bread will
fall. This causes the relative price of ale to rise, which in turn causes
the budget line to pivot through E to a new steeper slope. In other
words you need more bread to buy ale.
aH
b’G
a’H
aH
OH
b’H
bH
M
bG
bG
E
bH
V1
P’
1
V2
OG
a’G aG
aG
Competitive Equilibrium
Q: Will the price stop changing and reach equilibrium?
A: The relative price tends to change if there is excess
demand or excess supply for a good. It stops changing
and reaches equilibrium when the quantity demanded
equals the quantity supplied for all goods in the market.
We call this the equilibrium price, or market clearing
price.
• In the above diagram P’ is the equilibrium price and M
is the competitive allocation.
• At the equilibrium price the quantity of ale offered for
sale equals the quantity of ale bought and the quantity
of bread offered for sale equals the quantity of bread
bought .
Competitive Equilibrium
Q: Does an equilibrium price exist for economies
with more than two commodities?
A: There are a number of existence theorems in
economics that show that there exists a market
equilibrium price vector that clears the market. It
may seem scarcely possible but it is straightforward
to show that a market clearing price vector exists
for any competitive market with an arbitrarily large
set of commodities being traded. See a graduate
text for further details.
Competitive Equilibrium
Q: Will a market automatically adjust prices to reach equilibrium?
A: In economic models of 2 or more commodities, the ability of a
market to automatically adjust to equilibrium is examined in
uniqueness and stability theorems.
• The uniqueness and stability of an equilibrium price vector depends
on the shape of the demand and supply curves.
• The text (chapter 2) shows how there can be more than one market
price if the demand curve crosses the supply curve at more than
one point.
• If there is more than one equilibrium price, each equilibrium price
should be checked to see if it is a stable equilibrium. You are
responsible for this material.
• In general equilibrium theory, there are number of theorems which
identify the conditions under which an economy with n
commodities has a unique and stable equilibrium price. See a
graduate text for further details.
Competitive Equilibrium
Q: If there is a unique and stable market equilibrium price, do the
market trades coordinated by this price lead to an efficient (Pareto
optimal) allocation?
A: While this will not hold in all economies, it is easy to show that it
holds in a 2x2 competitive exchange economy.
1st Welfare Theorem: Every competitive allocation is Pareto optimal.
Proof: Utility maximization causes each person’s choice to be tangent
to the budget line defined by the market equilibrium price. If the
market is in equilibrium, there cannot be any excess demand or excess
supply, so each must choose an allocation at which their indifference
curves are tangent to each other. The competitive allocation is
therefore Pareto optimal (efficient) because there is no lens.
Competitive Equilibrium
Q: Is the Laissez-Faire market allocation the best
allocation available?
A: No. The competitive allocation is just one of
many efficient allocations that are hypothetically
available to a society (given by the contract
curve). If efficiency and fairness both matter,
there are a huge number of alternative
allocations which are efficient and might be
judged by society to be fairer.
Competitive Equilibrium
Q: If we want a fair society, is the market mechanism no longer useful?
A: The second welfare theorem suggests that the market mechanism might
still be useful in reaching a fairer point on the contract curve than provided
under Laissez-Faire.
2nd Welfare Theorem: Each Pareto optimal allocation is the competitive allocation
under some distribution of the endowed goods.
Proof: To prove this theorem we must show that for any point “Z” on the
Contract Curve, there exists an initial endowment “X” at which a market price
would emerge that would cause markets to clear and cause trading to
automatically take the market participants to Z. This is not that hard to show.
First choose some Z on the contract curve. This allocation has the property
that the MRSG = MRSH . Draw a line through Z having a slope q = MRSG = MRSH
. An allocation X anywhere along this line will necessarily cause a price p=q to
spontaneously emerge, and trade will immediately cause the allocation to
shift from X to Z.
Competitive Equilibrium
In the following diagram an initial endowment E leads to a market
equilibrium price that generates a competitive allocation M. While M is
Pareto optimal, so is Z. Society may prefer Z to M because while they are
both efficient, Z is more equitable. But how might we get to Z. The
second welfare theorem says to draw a line through Z with the same
slope as the MRS at Z for G and H. Any initial endowment X on this line
will produce a market price equal to this MRS and automatically lead
society to reallocate to Z. The trick then is to figure out how to reallocate
initial endowments from E to X and let the market do the rest.
aH
OH
bH
Z
●
●M
●E
●
X
bG
OG
aG
Competitive Equilibrium
Q: How might an efficient and equitable allocation be reached?
A: The theorem seems to suggest we should attempt to engineer a
reallocation of endowments and then let the market take us to an
efficient point on the contract curve. If this point is not ideal, further
iterative adjustments in this reallocation in endowments will iteratively
lead us closer a fair and efficient allocation. Essentially, the theorem
argues that those who want to increase equity should restrict their
policies to reallocating purchasing power. The details of individual
consumption should be left to individual decisions coordinated by
market prices.
Q: How can a simple adjustment of endowments be accomplished?
A: This requires a tax and transfer mechanism. As you are well aware,
such taxes and transfers introduce deadweight losses (inefficiency). A
mechanism for simple lump-sum adjustments of endowments has not
been discovered yet.
Competitive Equilibrium
Q: Does this mean that the 2nd Welfare theorem provides no
useful insight?
A: While it is no doubt impossible to engineer fair and
efficient allocations, it seems sensible that policy aimed at
helping the poor should:
• be restricted to redistributing purchasing power.
• After purchasing power has been redistributed, we should
then leave the poor to make their own choices about what
allocation they want to consume given market prices.
Q: If we introduce more commodities and production into our
exchange economy, does the first welfare theorem survive?
A: Yes. Read chapter 3 (as an option 4) for next week.
Competitive Equilibrium
Q: What are the mathematical relationships underlying this model of a 2x2 pure exchange
economy?
A: Chapter 3:




Each person chooses to consume at the point on the budget line where an indifference
curve just touches the budget line as a tangent.
This is equivalent to finding the point on the budget line which has an indifference
curve that has exactly the same slope as the budget line.
The slope of the budge line is the relative price of ale. P = (Pa / Pb) .
The absolute value of the slope of the indifference curve is called the MRS.
u  u ( a, b)
u
u
du 
da 
db  0
a
b
db
u / a MU a
MRS  


da du o u / b MU b
Competitive Equilibrium

Therefore the individual will choose the point on the their budget line where:
P  MRS
Pa MU a

Pb MU b

Our 2x2 economy has two individuals. Since the other individual faces the same
prices, they will both choose a point on their respective budget lines where their MRS
equals the price ratio.
P  MRS G  MRS H
Pa MU aG MU aH


G
Pb MU b
MU bH
Competitive Equilibrium
• The above condition will hold for any price, as individuals will
automatically adjust their consumption choices so the P=MRS. The
above condition is not an equilibrium as most prices will cause an
excess supply for one good and an excess demand for the other
good.
• A competitive equilibrium is an allocation a=(aG , bG , aH , bH) and a
price p = (pa/pb) such that the allocation is feasible and the market
clears (no excess demand or supply), and P = MRSG = MRSH .
• The market clearing price can be found by calculating the excess
demand equations, and substituting excess demand equations into
the market clearing condition. See chapter 3 for details and
examples. You are responsible.
• The competitive market equilibrium allocation is also Pareto
optimal as the allocation is feasible and MRSG = MRSH .
Competitive Equilibrium
End