2x2 Exchange Economy
ECON 3328A
Readings: Leach (2004), chapters 2 and 3
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 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).
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
a  aG  a H
Ale Endowment:
Bread Endowment: b  bG  bH
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
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
aH
OH
bG
A
E
bG
OG
bH
aG
aG
bH
Q: Why might society seek to choose a different allocation than provided for by the
endowment?
A: The endowment may be viewed as unfair and most certainly is inefficient.
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.
aH
OH
E
bG
U1
bH
V1
OG
aG
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.
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
bH
V1
OG
aG
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. 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).
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
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
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
aH
aH
U1
OH
U2
P
1
bH
bG
bG
E
V1
V2
OG
aG
bH
aG
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 a H  a H 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 .
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
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, 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 aG  aG equals the
quantity of ale bought a H  aH and the quantity of bread offered for sale bH  bH equals
the quantity of bread bought bG  bG .
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.
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.
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.
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. 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.
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.
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
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.
Q: Does this mean that the 2nd Welfare theorem has no useful insight?
A: While it is not doubt impossible to engineer a fair and efficient allocations, it seems
sensible that policy aimed at helping the poor should be restricted to redistribute
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 chapters 3 and 4 for next week.
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

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


Pb MU bG MU bH




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 MRS G  MRS H .