3.1 THE 22 EXCHANGE ECONOMY - Essential Microeconomicsessentialmicroeconomics.com/ChapterY3/SlideChapter3-1.pdf · Pareto-efficient allocations An allocation {}ˆ 1 hH x h = is Pareto

Essential Microeconomics -1-

© John Riley October 4, 2012

3.1 THE 2 2× EXCHANGE ECONOMY

Private goods economy 2

Pareto efficient allocations 3

Edgeworth box analysis 6

Market clearing prices and Walras’ Law 14

Walrasian Equilibrium 16

Equilibrium and Efficiency 22



Private goods exchange economy

Consumer (or household) , 1,...,h h H= has strictly increasing preferences h over h nX += .

We assume that the basic preference axioms are satisfied so that these are represented by a continuous

utility function ( )hU ⋅ .

Where it is helpful we will assume that U is continuously differentiable ( 1( )U ⋅ ∈ ).

Endowments: The initial allocation of commodities is 1{ }h Hhω = .

Feasible allocations: The final allocation 1{ }h Hhx = is feasible if

1 1

H Hh h

h hx ω

= =

≤∑ ∑



Pareto-efficient allocations

An allocation 1ˆ{ }h Hhx = is Pareto efficient if there is no other feasible allocation in which at least one

consumer is strictly better off and no consumer is worse off.

Consider an alternative allocation 1{ }h Hhx = in which consumers 2,…,H are all at least as well off.

That is, ˆˆ( ) ( )h h h h hU x U x U≥ ≡

Since 1ˆ{ }h Hhx = it cannot be the case that 1 1 1 1ˆ( ) ( )U x U x> .

Therefore 1 1 1 1ˆ( ) ( )U x U x≤ and so 1ˆ{ }h Hhx = solves the following maximization problem.

1 1 1

{ } 1 1

ˆˆ arg { ( ) | ( ) , 2,..., , }h

H Hh h h h h

x h hx Max U x U x U h H x ω

= =

= ≥ = ≤∑ ∑



Two commodity 2 consumer case (Alex and Bev)

For the special 2 2× case, Alex and Bev must

share the aggregate endowment 1 2( , )ω ω ω= .

Let ˆBx be the allocation to Bev and let B̂ be the set

of allocations that Bev prefers over ˆBx .

This is depicted in Figure 3.1-2. For any ˆBx B∈ ,

the allocation to Alex is A Bx xω= − . Thus the best

possible allocation to Alex that leaves Bev no worse

off is Alex’s utility maximizing allocation in B̂ .

Figure 3.1-2: Bev’s upper contour set



Edgeworth box diagram

Since preferences are strictly increasing

a PE allocation uses all the endowment A B A Bx x ω ω ω+ = = +

In the diagram the sum of the two consumption

vectors is the vector 1 2( , )ω ω , that is, the right

hand corner of the Edgeworth box.

Figure 3.1-3: Edgeworth box Diagram



For Pareto-efficiency, there can be no mutually

preferred alternative. One PE allocation is

depicted in Figure 3.1-4. As long as an allocation

ˆ ˆ ˆ{ , }A B Ax x xω= −

is in the interior of the Edgeworth box, a necessary

condition for the allocation to be PE is that the

slopes of the two indifference curves must be

equal. Thus the graph of the PE allocations is the set

of allocations to Alex (and hence Bev) satisfying

1 1

2 2

ˆ ˆ( ) ( )ˆ( )

ˆ ˆ( ) ( )

A BA B

A AA B

A B

U Ux xx xMRS x

U Ux xx x

∂ ∂∂ ∂

= =∂ ∂∂ ∂

, where ˆ ˆA Bx x ω+ = .

Figure 3.1-4: PE allocations



Example: Identical CES Preferences

If preferences are identical and CES with an

elasticity of substitution σ , the MRS for Alex

and Bev are 1/

2

1

( )h

h hh

xMRS x kx

σ

=

.

For a PE allocation in the interior of the Edgeworth

box, the indifference curves of the two consumers

must have the same slope, that is, 1/ 1/

2 2

1 1

A B

A B

x xx x

σ σ

=

hence 2 2

1 1

A B

A B

x xx x

= .

Figure 3.1-4: PE allocations with identical CES preferences



*

Example: Identical CES Preferences

If preferences are identical and CES with an

elasticity of substitution σ , the MRS for Alex

and Bev are 1/

2

1

( )h

h hh

xMRS x kx

σ

=

.

For a PE allocation in the interior of the Edgeworth

box, the indifference curves of the two consumers

must have the same slope, that is, 1/ 1/

2 2

1 1

A B

A B

x xx x

σ σ

=

hence 2 2

1 1

A B

A B

x xx x

= .

Ratio Rule: 1 1 1 1

2 2 2 2

a b a ba b a b

+= =

+

Proof: If 1 1

2 2

a b ka b

= = then 1 2a ka= and 1 2b kb= and so 1 1 2 2( )a b k a b+ = + .

Hence 1 1

2 2

a b ka b+

=+

.




Appealing to the Ratio Rule and then setting

demand equal to supply,

2 2 2 2 2

1 1 1 1

A B A B

A B A Ba

x x x xx x x x

ωω

+= = =

+.

Thus, in a PE allocation each consumer is

allocated a fraction of the aggregate endowment.

It follows that for each consumer the marginal

rate of substitution is 1/

2

1

ˆ( )h hMRS x kσ

ωω

=

. (3.1-1)

The PE allocations are depicted in Figure 3.1-4.




Walrasian Equilibrium for an Exchange Economy

Let 0p ≥ be a price vector of this exchange economy.

In a WE each consumer is a price taker.

We write the set of consumers as {1,..., }HH = .

***







We assume that preferences are strictly convex so consumer h has is a unique most preferred

consumption vector, ( , )h hx p ω .

( , )h hx p ω solves { ( ) | }h h

xMax U x p x p ω⋅ ≤ ⋅ .

**








consumption vectr, ( , )h hx p ω .

( , )h hx p ω solves { ( ) | }h h


Total endowment vector: h

hω ω

∈

= ∑H

Total or “market” demand: ( ) ( , )h h

hx p x p ω

∈

= ∑H

Excess demand: ( ) ( )z p x p ω= − .

*








consumption vector, ( , )h hx p ω .

( , )h hx p ω solves { ( ) | }h h


Total endowment vector: h

hω ω

∈

= ∑H

Total or “market” demand: ( ) ( , )h h

hx p x p ω

∈

= ∑H

Excess demand: ( ) ( )z p x p ω= − .

Definition: Market Clearing Prices

Let ( )jz p be the excess demand for commodity j at the price vector 0p ≥ . The market for commodity

j clears if ( ) 0jz p ≤ and ( ) 0j jp z p = .



Walras’ Law: If preferences satisfy local non-satiation and all markets but one clear then the

remaining market must also clear.

If preferences satisfy local non-satiation, then a consumer must spend all of his income.

Why is this?

**






Why is this?

Then for any price vector p the market value of excess demands must be zero.

( ) ( ) ( ( )) ( )h h h h

h hp z p p x p x p x pω ω ω

∈ ∈

⋅ = ⋅ − = ⋅ − = ⋅ − ⋅∑ ∑H H

.

Because all consumers spend their entire wealth the right hand expression is zero. Hence

1

( ) ( ) ( ) 0n

i i j jjj i

p z p p z p p z p=≠

⋅ = + =∑ .

*






Why is this?

Then for any price vector p the market value of excess demands must be zero.

( ) ( ) ( ( )) ( )h h h h

h hp z p p x p x p x pω ω ω

∈ ∈

⋅ = ⋅ − = ⋅ − = ⋅ − ⋅∑ ∑H H

.

Because all consumers spend their entire wealth the right hand expression is zero. Hence

1

( ) ( ) ( ) 0n

i i j jjj i

p z p p z p p z p=≠

⋅ = + =∑ .

Therefore if all markets but market i clear then market i must clear as well.

Definition: Walrasian Equilibrium price vector

The price vector 0p > is a WE price vector if all markets clear.



Edgeworth box example

In a Walrasian equilibrium consumers choose

the best point in their budget sets given a price

vector 1 2( , )p p p= .

In the figure these are the dotted orange

and green triangles.

**

Figure 3.1-5: Excess supply of commodity 1

N






vector 1 2( , )p p p= .



It is very important to note that consumers consider

only their budget sets. In the case depicted, both of

these budget sets extend beyond the boundary of the

Edgeworth box (the set of feasible allocations).

*


N






vector 1 2( , )p p p= .



It is very important to note that consumers consider

only their budget sets. In the case depicted, both of

these budget sets extend beyond the boundary of the

Edgeworth box (the set of feasible allocations).

The heavily shaded triangles indicate the desired trades

of the two consumers. As depicted, Alex

wants to trade from the endowment point N to his most preferred desired consumption Ax , whereas

Bev wishes to trade from N to Bx . Thus, there is excess supply of commodity 1.


N



By lowering the price of commodity 1 (relative to

commodity 2) the budget line becomes less steep

until eventually supply equals demand. The

Walrasian equilibrium E is depicted in Figure 3.1-6.

Figure 3.1-6: Walrasian equilibrium

N



Class Exercise: Which (if any) of these figures depicts a Walrasian equilibrium?

In the left figure the budget line is tangential to Bev’s indifference curve at ˆ Ax .

In the middle figure slope of the budget lies between the slopes of the two consumers at ˆ Ax .

In the right figure the budget line is tangential to Alex’s indifference curve at ˆ Ax .



Equilibrium and Efficiency

In Figure 3.1-6 the WE allocation is in the interior of the

Edgeworth box. Thus the marginal rates of substitution

must both be equal to the price ratio:

1 1 1

2

2 2

( ) ( )( ) ( )

( ) ( )

A BA B

A A B BA B

A B

U Ux xx p xMRS x MRS x

U Upx xx x

∂ ∂∂ ∂

= = = =∂ ∂∂ ∂

Since the MRS are equal, it follows that the

WE allocation must be PE.

Figure 3.1-6: Walrasian equilibrium

N

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3.1 THE 22 EXCHANGE ECONOMY - Essential Microeconomicsessentialmicroeconomics.com/ChapterY3/SlideChapter3-1.pdf · Pareto-efficient allocations An allocation {}ˆ 1 hH x h = is Pareto