3.1 THE 22 EXCHANGE ECONOMY - Essential...
Transcript of 3.1 THE 22 EXCHANGE ECONOMY - Essential...
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
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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
= =
≤∑ ∑
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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
= =
= ≥ = ≤∑ ∑
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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
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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
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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
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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
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*
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+
=+
.
Figure 3.1-4: PE allocations with identical CES preferences
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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.
Figure 3.1-4: PE allocations with identical CES preferences
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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 = .
***
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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⋅ ≤ ⋅ .
**
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© John Riley October 4, 2012
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 vectr, ( , )h hx p ω .
( , )h hx p ω solves { ( ) | }h h
xMax U x p x p ω⋅ ≤ ⋅ .
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= − .
*
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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⋅ ≤ ⋅ .
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 = .
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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?
**
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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?
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=≠
⋅ = + =∑ .
*
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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?
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.
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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
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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.
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).
*
Figure 3.1-5: Excess supply of commodity 1
N
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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.
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.
Figure 3.1-5: Excess supply of commodity 1
N
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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
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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 .
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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