Chapter 32: Exchange
The prices of goods affect people’s demand and supply for a particular good. Certainly the prices of
substitutes and complements for a good will influence the demand for it, and, more subtly, the prices of
goods that people sell will affect the amount of income they have and thereby influence how much of other
goods they will be able to buy.
32.1 The Edgeworth Box
This tool can be used to analyse the exchange of two goods between two people. The Edgeworth box allows
us to depict the endowments and preferences of two individuals in one convenient diagram.
Example: Two people (A,B) and two goods (1,2)
● A’s consumption bundle: XA = (x1A, x2A )
○ x1A represents A’s consumption of good 1
○ x2A represents A’s consumption of good 2
● B’s consumption bundle: XB = (x1B, x2B )
○ x1B represents B’s consumption of good 1
○ x2B represents B’s consumption of good 2
A pair of consumption bundles, XA and XB , is called an allocation. An allocation is a feasible allocation if the
total amount of each good consumed is equal to the total amount available:
A particular feasible allocation that is of interest is the initial endowment allocation, (ω1A, ω2A) and (ω1B,
ω2B). This is the allocation that the consumers start with. It consists of the amount of each good that
consumers bring to the market. They will exchange some of these goods with each other in the course of
trade to end up at a final allocation.
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,If we start at A’s origin in the lower left-hand corner and move up and to the right, we will be moving to
allocations that are more preferred by A. As we move down and to the left we will be moving to allocations
that are more preferred by B.
32.2 Trade
Point W in Figure 32.1:
● The region where A is better off than at her endowment consists of all the bundles above her
indifference curve through W. The region where B is better off than at his endowment consists of all
the allocations that are above—from his point of view—his indifference curve through W.
● Both are better off at the intersection of these two points.
● Point M is mutually advantageous
● Person A gives up [x1A − ω1A] units of good 1 and acquires in exchange [x2B − ω2B] units of good 2
● This means that B acquires [x1A − ω1A] units of good 1 and gives up [x2B − ω2B] units of good 2.
● Any allocation inside the lens-shaped region would be possible
32.3 Pareto Efficient Allocations
A Pareto efficient allocation can be described as an allocation where:
1. There is no way to make all the people involved better off
2. There is no way to make some individual better off without making someone else worse off
3. All of the gains from trade have been exhausted
4. There are no mutually advantageous trades to be made
The indifference curves of the two agents must be tangent at any Pareto efficient. If they cross then there
must be some mutually advantageous trade—so that point cannot be Pareto efficient.
• To find the Pareto efficient allocations, we move along A’s indifference curve until you find the point
that is the best point for B. This will be a Pareto efficient point, and thus both indifference curves
must be tangent at this point.
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,The set of all Pareto efficient points in the Edgeworth box is known as the Pareto set, or the contract curve.
In a typical case the contract curve will stretch from A’s origin to B’s origin across the Edgeworth box, as
shown in Figure 32.2. If we start at A’s origin, A has none of either good and B holds everything. This is Pareto
efficient since the only way A can be made better off is to take something away from B. As we move up the
contract curve A is getting more and more well-off until we finally get to B’s origin.
32.4 Market Trade
Suppose that we have a third party who is willing to act as an “auctioneer” for the two agents A and B. The
auctioneer chooses a price for good 1 and a price for good 2 and presents these prices to the agents A and B.
Each agent then sees how much his or her endowment is worth at the prices (p 1,p2) and decides how much of
each good he or she would want to buy at those prices.
➢ Note: if there are really only two people involved in the transaction, they would probably attempt to
bargain over the terms of trade.
The gross demand of agent A for good 1, say, is the total amount of good 1 that he wants at the going prices.
The net/excess demand of agent A for good 1 is the difference between this total demand and the initial
endowment of good 1 that agent A holds. This is denoted by e1A .
e1A = x1A - w1A
For arbitrary prices (p1,p2) there is no guarantee that supply will equal demand—in either sense of demand.
In terms of net demand, this means that the amount that A wants to buy (or sell) will not necessarily equal
the amount that B wants to sell (or buy). In terms of gross demand, this means that the total amount that the
two agents want to hold of the goods is not equal to the total amount of that goods available.
➢ We say that in this case the market is in disequilibrium.
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, When the amount that A wants to buy of good 1 just equals the amount that B wants to sell of good 1, and
similarly for good 2, the market is in equilibrium.
➢ This is also called a competitive equilibrium or a Walrasian equilibrium
➢ Each of these terms refers to the same thing: a set of prices such that each consumer is choosing his
or her most-preferred affordable bundle, and all consumers’ choices are compatible in the sense that
demand equals supply in every market.
We know that if each agent is choosing the best bundle that he can afford, then his marginal rate of
substitution between the two goods must be equal to the ratio of the prices. But if all consumers are facing
the same prices, then all consumers will have to have the same marginal rate of substitution between each of
the two goods. In terms of Figure 32.4, an equilibrium has the property that each agent’s indifference curve is
tangent to his budget line. But since each agent’s budget line has the slope −p1/p2, this means that the two
agents’ indifference curves must be tangent to each other.
32.5 The Algebra of Equilibrium
If we let x1A(p1,p2) be agent A’s demand function for good 1 and x1B(p1,p2) be agent B’s demand function for
good 1, and define the analogous expressions for good 2, we can describe this equilibrium as a set of prices
(p1*,p2*) such that:
These equations say that in equilibrium the total demand for each good should be equal to the total supply.
Another way to describe the equilibrium is to rearrange these two equations to get:
These equations say that the sum of net demands of each agent for each good should be zero.
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The prices of goods affect people’s demand and supply for a particular good. Certainly the prices of
substitutes and complements for a good will influence the demand for it, and, more subtly, the prices of
goods that people sell will affect the amount of income they have and thereby influence how much of other
goods they will be able to buy.
32.1 The Edgeworth Box
This tool can be used to analyse the exchange of two goods between two people. The Edgeworth box allows
us to depict the endowments and preferences of two individuals in one convenient diagram.
Example: Two people (A,B) and two goods (1,2)
● A’s consumption bundle: XA = (x1A, x2A )
○ x1A represents A’s consumption of good 1
○ x2A represents A’s consumption of good 2
● B’s consumption bundle: XB = (x1B, x2B )
○ x1B represents B’s consumption of good 1
○ x2B represents B’s consumption of good 2
A pair of consumption bundles, XA and XB , is called an allocation. An allocation is a feasible allocation if the
total amount of each good consumed is equal to the total amount available:
A particular feasible allocation that is of interest is the initial endowment allocation, (ω1A, ω2A) and (ω1B,
ω2B). This is the allocation that the consumers start with. It consists of the amount of each good that
consumers bring to the market. They will exchange some of these goods with each other in the course of
trade to end up at a final allocation.
1 of 77
,If we start at A’s origin in the lower left-hand corner and move up and to the right, we will be moving to
allocations that are more preferred by A. As we move down and to the left we will be moving to allocations
that are more preferred by B.
32.2 Trade
Point W in Figure 32.1:
● The region where A is better off than at her endowment consists of all the bundles above her
indifference curve through W. The region where B is better off than at his endowment consists of all
the allocations that are above—from his point of view—his indifference curve through W.
● Both are better off at the intersection of these two points.
● Point M is mutually advantageous
● Person A gives up [x1A − ω1A] units of good 1 and acquires in exchange [x2B − ω2B] units of good 2
● This means that B acquires [x1A − ω1A] units of good 1 and gives up [x2B − ω2B] units of good 2.
● Any allocation inside the lens-shaped region would be possible
32.3 Pareto Efficient Allocations
A Pareto efficient allocation can be described as an allocation where:
1. There is no way to make all the people involved better off
2. There is no way to make some individual better off without making someone else worse off
3. All of the gains from trade have been exhausted
4. There are no mutually advantageous trades to be made
The indifference curves of the two agents must be tangent at any Pareto efficient. If they cross then there
must be some mutually advantageous trade—so that point cannot be Pareto efficient.
• To find the Pareto efficient allocations, we move along A’s indifference curve until you find the point
that is the best point for B. This will be a Pareto efficient point, and thus both indifference curves
must be tangent at this point.
2 of 77
,The set of all Pareto efficient points in the Edgeworth box is known as the Pareto set, or the contract curve.
In a typical case the contract curve will stretch from A’s origin to B’s origin across the Edgeworth box, as
shown in Figure 32.2. If we start at A’s origin, A has none of either good and B holds everything. This is Pareto
efficient since the only way A can be made better off is to take something away from B. As we move up the
contract curve A is getting more and more well-off until we finally get to B’s origin.
32.4 Market Trade
Suppose that we have a third party who is willing to act as an “auctioneer” for the two agents A and B. The
auctioneer chooses a price for good 1 and a price for good 2 and presents these prices to the agents A and B.
Each agent then sees how much his or her endowment is worth at the prices (p 1,p2) and decides how much of
each good he or she would want to buy at those prices.
➢ Note: if there are really only two people involved in the transaction, they would probably attempt to
bargain over the terms of trade.
The gross demand of agent A for good 1, say, is the total amount of good 1 that he wants at the going prices.
The net/excess demand of agent A for good 1 is the difference between this total demand and the initial
endowment of good 1 that agent A holds. This is denoted by e1A .
e1A = x1A - w1A
For arbitrary prices (p1,p2) there is no guarantee that supply will equal demand—in either sense of demand.
In terms of net demand, this means that the amount that A wants to buy (or sell) will not necessarily equal
the amount that B wants to sell (or buy). In terms of gross demand, this means that the total amount that the
two agents want to hold of the goods is not equal to the total amount of that goods available.
➢ We say that in this case the market is in disequilibrium.
3 of 77
, When the amount that A wants to buy of good 1 just equals the amount that B wants to sell of good 1, and
similarly for good 2, the market is in equilibrium.
➢ This is also called a competitive equilibrium or a Walrasian equilibrium
➢ Each of these terms refers to the same thing: a set of prices such that each consumer is choosing his
or her most-preferred affordable bundle, and all consumers’ choices are compatible in the sense that
demand equals supply in every market.
We know that if each agent is choosing the best bundle that he can afford, then his marginal rate of
substitution between the two goods must be equal to the ratio of the prices. But if all consumers are facing
the same prices, then all consumers will have to have the same marginal rate of substitution between each of
the two goods. In terms of Figure 32.4, an equilibrium has the property that each agent’s indifference curve is
tangent to his budget line. But since each agent’s budget line has the slope −p1/p2, this means that the two
agents’ indifference curves must be tangent to each other.
32.5 The Algebra of Equilibrium
If we let x1A(p1,p2) be agent A’s demand function for good 1 and x1B(p1,p2) be agent B’s demand function for
good 1, and define the analogous expressions for good 2, we can describe this equilibrium as a set of prices
(p1*,p2*) such that:
These equations say that in equilibrium the total demand for each good should be equal to the total supply.
Another way to describe the equilibrium is to rearrange these two equations to get:
These equations say that the sum of net demands of each agent for each good should be zero.
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