Exchange Economies

Endowments

• Revisit the consumer optimization problem
• The consumer starts with no income: i.e. set m=0
• The consumer starts with a given endowment instead, i.e. a starting bundle
• Definition: The amount of resources that the consumer starts with is called his initial endowment
• We will denote the initial endowment by ω = (ω₁, ω₂)
• Ex: ω = (10, 2) means that the consumer starts with 10 units of good 1 and 2 units of good 2

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Endowments

• For which bundles could we exchange this endowment?
• i.e. what is the budget set?
• Ex: suppose that ω = (10, 2) and p₁=2, p₂ = 3
• Then the consumer can sell her endowment for 10*2+2*3 = 26
• She can now purchase any bundle (x₁,x₂) such that 2x₁+3x₂ ≤ 26

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• In general:

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• p₁x₁ + p₂x₂ ≤ p₁ ω₁ + p₂ ω₂

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Budget Constraints Revisited

x₂

x₁

ω₁

ω₂

Budget Constraints Revisited

x₂

x₁

ω₁

ω₂

Budget set

Budget Constraints Revisited

x₂

x₁

ω₁

ω₂

Budget Constraints Revisited

x₂

x₁

ω₁

ω₂

Budget set

Budget Constraints Revisited

x₂

x₁

ω₁

ω₂

The endowment point is always on �the budget constraint.

Budget Constraints Revisited

x₂

x₁

ω₁

ω₂

The endowment point is always on �the budget constraint.

So price changes pivot the�constraint around the� endowment point.

Net Demand

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• Suppose that the optimal bundle under the constraint p₁x₁ + p₂x₂ ≤ p₁ ω₁ + p₂ ω₂ is (x₁*,x₂*)
• Then p₁x₁^* + p₂x₂^* = p₁ ω₁ + p₂ ω₂
• Rewrite this as:
• p₁(x₁^*- ω₁) + p₂(x₂^*- ω₂) = 0
• x₁^*- ω₁ is the net demand for good 1
• i.e the sum of the values of the consumer’s net demands equals zero

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Net Demands

• Suppose ω = (10, 2) and p₁=2, p₂=3. Then the constraint is �p₁x₁ + p₂x₂ ≤ p₁ ω₁ + p₂ ω₂ = 26

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• If the consumer demands (x₁*,x₂*) = (7,4). Net demands are: �x₁*- ω₁ = 7-10 = -3, i.e. the value of net demand for good 1 is -6�x₂*- ω₂ = 4 - 2 = +2, i.e. the value of net demand for good 2 is +6

Net Demands

x₂

x₁

ω₁

ω₂

x₂*

x₁*

At prices (p₁,p₂) the consumer�sells units of good 1 to acquire�more units of good 2.

Net Demands

x₂

x₁

ω₁

ω₂

x₂*

x₁*

At prices (p₁’,p₂’) the consumer�sells units of good 2 to acquire�more of good 1.

Net Demands

x₂

x₁

x₂*=ω₂

x₁*=ω₁

At prices (p₁”,p₂”) the consumer�consumes her endowment; net�demands are all zero.

Exchange

• Two consumers, A and B
• Their endowments are:
• ω^A = (ω^A ₁, ω^A ₂) and ω^B = (ω^B ₁, ω^B ₂)
• Ex: ω^A = (6,4) and ω^B = (2,2)
• The total quantities of both goods available are:
• ω^A ₁+ ω^B₁ = 6 + 2 = 8 units of good 1
• ω^A ₂+ ω^B₂ = 4 + 2 = 6 units of good 2
• Edgeworth and Bowley devised a diagram, called an Edgeworth box, to show all possible allocations of the available quantities of goods 1 and 2 between the two consumers

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Starting an Edgeworth Box

Starting an Edgeworth Box

Width =

Starting an Edgeworth Box

Width =

Height =

Starting an Edgeworth Box

Width =

Height =

The dimensions of�the box are the�quantities available�of the goods.

Feasible Allocations

• The purpose of the Edgeworth Box is to represent all Feasible Allocations
• Feasible Allocation: A pair of consumption bundles, x^A = (x^A ₁, x^A ₂) and x^B = (x^B ₁, x^B ₂) such that
• x^A ₁+ x^B ₁ = ω ^A ₁+ ω ^B ₁
• x^A ₂+ x^B ₂ = ω ^A ₂+ ω ^B ₂
• i.e. all ways in which we can divide the total endowment in this economy between the two consumers
• One such allocation is the initial endowment:�

The Endowment Allocation

Width =

Height =

The endowment�allocation is

and

The Endowment Allocation

Width =

Height =

O_A

O_B

6

8

O_A

O_B

6

8

4

6

O_A

O_B

6

8

4

6

2

2

O_A

O_B

6

8

4

6

2

2

The�endowment�allocation

O_A

O_B

The�endowment�allocation

O_A

O_B

O_A

O_B

Feasible Reallocations

• All points in the box, including the boundary, represent feasible allocations of the combined endowments.

Feasible Reallocations

• All points in the box, including the boundary, represent feasible allocations of the combined endowments.
• Which allocations will be blocked by one or both consumers?
• Which allocations make both consumers better off?

O_A

For consumer A.

Adding Preferences to the Box

More preferred

For consumer A.

O_A

For consumer B.

O_B

More preferred

For consumer B.

O_B

More preferred

For consumer B.

O_B

O_A

O_B

Pareto Efficiency

• We say that a social outcome X is a Pareto Improvement over a social outcome Y if everyone weakly prefers X to Y and at least one person strictly prefers X to Y. Formally, if there are i=1,…,n people:
• X ≿_i Y for all i=1,…,n, and
• X ≻_j Y for at least one person j
• i.e. Y makes at least one person strictly better off without making anyone else worse off
• We also say that X Pareto Dominates Y

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• An outcome X is Pareto Efficient (Pareto Optimal) if it is not Pareto Dominated by any other outcome
• i.e. if the only way to make anyone better off would come at the cost of hurting someone else

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O_A

O_B

O_A

O_B

The set of Pareto-�improving allocations

Pareto-Improvements

• Since each consumer can refuse to trade, the only possible outcomes from exchange are Pareto-improving allocations.
• But which particular Pareto-improving allocation will be the outcome of trade?

O_A

O_B

The set of Pareto-�improving reallocations

Pareto-Improvements

Trade�improves both�A’s and B’s welfares.�This is a Pareto-improvement�over the endowment allocation.

New mutual gains-to-trade region� is the set of all further Pareto-� improving� reallocations.�

Trade�improves both�A’s and B’s welfares.�This is a Pareto-improvement�over the endowment allocation.

Further trade cannot improve� both A and B’s� welfares.

Pareto Optimal Allocation

• An allocation is Pareto Optimal if there is no further Pareto Improvement
• i.e. if the only way to make anyone better off is by hurting someone else

Pareto-Optimality

Better for�consumer B

Better for�consumer A

Pareto-Optimality

A is strictly better off� but B is strictly worse� off

Pareto-Optimality

A is strictly better off� but B is strictly worse� off

B is strictly better�off but A is strictly�worse off

A is strictly better off� but B is strictly worse� off

B is strictly better�off but A is strictly�worse off

Both A and�B are worse�off

A is strictly better off� but B is strictly worse� off

B is strictly better�off but A is strictly�worse off

Both A�and B are� worse� off

Both A and�B are worse�off

Pareto-Optimality

The allocation is�Pareto-optimal since the�only way one consumer’s

welfare can be increased is to�decrease the welfare of the other�consumer.

Pareto-Optimality

• Where are all of the Pareto-optimal allocations of the endowment?

O_A

O_B

All the allocations marked by�a are Pareto-optimal.

Pareto-Optimality

• The contract curve is the set of all Pareto-optimal allocations.

O_A

O_B

All the allocations marked by�a are Pareto-optimal.

The contract curve

Pareto-Optimality

• But to which of the many allocations on the contract curve will consumers trade?
• That depends upon how trade is conducted.
• In perfectly competitive markets? By one-on-one bargaining?

O_A

O_B

The set of Pareto-�improving reallocations

O_A

O_B

O_A

O_B

Pareto-optimal trades blocked� by B

Pareto-optimal trades blocked� by A

O_A

O_B

Pareto-optimal trades not blocked� by A or B

O_A

O_B

Pareto-optimal trades not blocked� by A or B are the core.

The Core

• The core is the set of all Pareto-optimal allocations that are welfare-improving for both consumers relative to their own endowments (i.e. Pareto Efficient Allocations that are not blocked by either consumer).
• Rational trade should achieve a core allocation.

Trade in Competitive Markets

• So far we have seen that without any specific assumptions on how trade is accomplished:
• The only rational outcomes (allocations) must be in the Core
• All allocations in the Core are Pareto Efficient
• Now let us make a specific assumption on how trade is conducted
• Assume that markets are competitive: Each consumer is a price-taker trying to maximize her own utility given p₁, p₂ and her own endowment. �

Trade in Competitive Markets

O_A

For consumer A.

Trade in Competitive Markets

For consumer B.

O_B

General Equilibrium

• A general equilibrium occurs when
• (a) Each individual maximizes utility, and
• (b) equilibrium prices p₁ and p₂ cause both the markets for commodities 1 and 2 to clear; i.e.

and

Budget Constraints in Edgeworth Box

• Important observation: If, for any given prices p₁ and p_2, we draw the budget constraint for individual A in the Edgeworth box, then it would also be the budget constraint for individual B. This is true because:
• Each budget constraint has to pass through the endowment point (which is a single point in the Edgeworth box)
• And the price ratio is the same for both consumers since they are price takers (competitive market)�

Trade in Competitive Markets

O_A

O_B

O_A

O_B

Can this PO allocation be

achieved as the outcome of

a general equilibrium?

O_A

O_B

Budget constraint for consumer A

O_A

O_B

Budget constraint for consumer A

O_A

O_B

Budget constraint for consumer B

O_A

O_B

Budget constraint for consumer B

O_A

O_B

But

O_A

O_B

and

Trade in Competitive Markets

• So at the given prices p₁ and p₂ there is an
• excess supply of commodity 1
• excess demand for commodity 2.
• This will cause p₁ to decrease, while p₂ to increase
• i.e. prices are not stable
• This can’t be an equilibrium

O_A

O_B

Which PO allocations can be�achieved by competitive trading?

O_A

O_B

Which PO allocations can be�achieved by competitive trading?

O_A

O_B

Which PO allocations can be�achieved by competitive trading?

O_A

O_B

Budget constraint for consumer A

O_A

O_B

O_A

O_B

and

Competitive Equilibrium

• Does an equilibrium bundle exist?
• Yes, as long as:
• The demand functions of all individuals are continuous
• Preferences are strictly convex
• How do we solve for the equilibrium?
• What are we given?
• Initial endowment: (ω^A ₁, ω^A ₂) and (ω^B ₁, ω^B ₂)
• And preferences (utility functions)
• What are we trying to solve for?
• Equilibrium allocation (x*^A₁,x*^A₂); (x*^B₁,x*^B₂);
• Equilibrium Prices (p₁,p₂)

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Competitive Equilibrium

Welfare Theorems

• First Fundamental Theorem of Welfare Economics:
• Every Competitive Equilibrium is Pareto Efficient

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• Second Fundamental Theorem of Welfare Economics:
• Any Pareto Efficient outcome can be supported as a Competitive Equilibrium for some prices (as long as we can change the initial endowment)

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