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Essential Microeconomics -1-
© John Riley
3. THE EXCHANGE ECONOMY
Pareto efficient allocations 2
Edgeworth box analysis 5
Market clearing prices 13
Walrasian Equilibrium 16
Equilibrium and Efficiency 22
First welfare theorem 24
Second welfare theorem (convex, differentiable economy) 28
The homothetic preference 2 2× economy 41
Second welfare theorem (convex economy) 49
Essential Microeconomics -2-
© John Riley
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
= =
≤∑ ∑
*
Essential Microeconomics -3-
© John Riley
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≥ ≡
Then 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
= =
= ≥ = ≤∑ ∑
Essential Microeconomics -4-
© John Riley
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
Essential Microeconomics -5-
© John Riley
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
Essential Microeconomics -6-
© John Riley
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 with identical CES preferences
Essential Microeconomics -7-
© John Riley
Example: Identical CES Preferences
If preferences are CES with elasticity of
substitution σ , Alex and Bev have a 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
Essential Microeconomics -8-
© John Riley
Example: Identical CES Preferences
If preferences are CES with elasticity of
substitution σ , Alex and Bev have a 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
Essential Microeconomics -9-
© John Riley
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
Essential Microeconomics -10-
© John Riley
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= .
***
Essential Microeconomics -11-
© John Riley
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 .
( , ) arg { ( ) | }h h h h
xx p Max U x p x pω ω= ⋅ ≤ ⋅ .
**
Essential Microeconomics -12-
© John Riley
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 .
( , ) arg { ( ) | }h h h h
xx p Max U x p x pω 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= − .
*
Essential Microeconomics -13-
© John Riley
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 .
( , ) arg { ( ) | }h h h h
xx p Max U x p x pω 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 = .
Essential Microeconomics -14-
© John Riley
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?
**
Essential Microeconomics -15-
© John Riley
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=≠
⋅ = + =∑ .
*
Essential Microeconomics -16-
© John Riley
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.
Essential Microeconomics -17-
© John Riley
Edgeworth box example
In a Walrasian equilibrium consumers choose
the best point in their budget sest given a price
vector 1 2( , )p p p= .
In the figure these are the lightly shaded orange
and green triangles.
**
Figure 3.1-5: Excess supply of commodity 1
N
Essential Microeconomics -18-
© John Riley
Edgeworth box example
In a Walrasian equilibrium consumers choose
the best point in their budget set given a price
vector 1 2( , )p p p= .
In the figure these are the lightly shaded 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
Essential Microeconomics -19-
© John Riley
Edgeworth box example
In a Walrasian equilibrium consumers choose
the best point in their budget set given a price
vector 1 2( , )p p p= .
In the figure these are the lightly shaded 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
Essential Microeconomics -20-
© John Riley
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
Essential Microeconomics -21-
© John Riley
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 right-hand figure the budget line is tangential to Alex’s indifference curve.
Essential Microeconomics -22-
© John Riley
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
Essential Microeconomics -23-
© John Riley
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.
To prove that this result holds very generally, we will appeal to the Duality Lemma (Section 2.2). That
is, if the local non-satiation property holds, then the utility-maximizing bundle is cost minimizing
among all preferred consumption bundles.
Duality Lemma
arg { ( ) | }h
h h h h h
xx Max U x p x p ω= ⋅ ≤ ⋅ ⇒ { | ( ) ( )}
h
h h h h h h
xp x Min p x U x U x⋅ = ⋅ ≥ .
Figure 3.1-6: Walrasian equilibrium
N
Essential Microeconomics -24-
© John Riley
Proposition 3.1-2: First welfare theorem for an exchange economy
If preferences satisfy local non-satiation, a WE allocation in an exchange economy is PE.
Proof: Let { }hhx ∈H be a WE allocation for the exchange economy with endowments { }h
hω ∈H . Let
0p ≥ be the WE price vector.
Consider any allocation { }hhx ∈H that is Pareto-preferred to { }h
hx ∈H . Because none of the consumers
can be worse off in the Pareto-preferred allocation, it follows from the Duality Lemma that
0, .h hp x p x h⋅ − ⋅ ≥ ∈H
**
Essential Microeconomics -25-
© John Riley
Proposition 3.1-2: First welfare theorem for an exchange economy
If preferences satisfy local non-satiation, a WE allocation in an exchange economy is PE.
Proof: Let { }hhx ∈H be a WE allocation for the exchange economy with endowments { }h
hω ∈H . Let
0p ≥ be the WE price vector.
Consider any allocation { }hhx ∈H that is Pareto-preferred to { }h
hx ∈H . Because none of the consumers
can be worse off in the Pareto-preferred allocation, it follows from the Duality Lemma that
0, .h hp x p x h⋅ − ⋅ ≥ ∈H
Moreover at least one consumer must be strictly better off. Since hx is the most preferred allocation in
the budget set, it follows that
0h hp x p x⋅ − ⋅ > , for some h .
*
Essential Microeconomics -26-
© John Riley
Proposition 3.1-2: First welfare theorem for an exchange economy
If preferences satisfy local non-satiation, a WE allocation in an exchange economy is PE.
Proof: Let { }hhx ∈H be a WE allocation for the exchange economy with endowments { }h
hω ∈H . Let
0p ≥ be the WE price vector.
Consider any allocation { }hhx ∈H that is Pareto-preferred to { }h
hx ∈H . Because none of the consumers
can be worse off in the Pareto-preferred allocation, it follows from the Duality Lemma that
0, .h hp x p x h⋅ − ⋅ ≥ ∈H
Moreover at least one consumer must be strictly better off. Since hx is the most preferred allocation in
the budget set, it follows that
0h hp x p x⋅ − ⋅ > , for some h .
Summing over consumers,
( ) 0h h
h hp x x
∈ ∈
⋅ − >∑ ∑H H
.
Also all markets clear in a Walrasian equilibrium. Therefore
( ) 0h h
h hp x p ω
∈ ∈
⋅ − ⋅ =∑ ∑H H
.
Essential Microeconomics -27-
© John Riley
Combining these results,
0h h
h hp x ω
∈ ∈
⎛ ⎞⋅ − >⎜ ⎟⎝ ⎠∑ ∑H H
Because 0p ≥ , it follows that there must be some commodity j such that 0h hj j
h hx ω
∈ ∈
− >∑ ∑H H
. Thus
all Pareto-preferred allocations are infeasible.
Q.E.D.
Essential Microeconomics -28-
© John Riley
Second Welfare Theorem
We now argue that, as long as preferences are convex,
any PE allocation is also a WE allocation for some
redistribution of resources.
Consider the PE allocation ˆ ˆ,A Bx x where ˆ ˆA Bx x ω+ = in Figure 3.1-7. The shaded regions are the allocations
where either Alex or Bev is better off.
**
Figure 3.1-7: PE allocation
Essential Microeconomics -29-
© John Riley
Second Welfare Theorem
We now argue that, as long as preferences are convex,
any PE allocation is also a WE allocation for some
redistribution of resources.
Consider the PE allocation ˆ ˆ,A Bx x where ˆ ˆA Bx x ω+ = in Figure 3.1-7. The shaded regions are the allocations
where either Alex or Bev is better off.
If preferences are convex, each of these sets is convex so,
by the Separating Hyperplane Theorem, we can draw a line
ˆA Ap x p x⋅ = ⋅ through ˆ Ax separating the two sets.
*
Figure 3.1-7: PE allocation
Essential Microeconomics -30-
© John Riley
Second Welfare Theorem
We now argue that, as long as preferences are convex,
any PE allocation is also a WE allocation for some
redistribution of resources.
Consider the PE allocation ˆ ˆ,A Bx x where ˆ ˆA Bx x ω+ = in Figure 3.1-7. The shaded regions are the allocations
where either Alex or Bev is better off.
If preferences are convex, each of these sets is convex so,
by the Separating Hyperplane Theorem, we can draw a line
ˆA Ap x p x⋅ = ⋅ through ˆ Ax separating the two sets.
If the endowments are ˆ ˆ ,h hx hω = ∈H each individual maximizes by choosing his or her endowment.
Because demand equals supply for each individual, all markets clear. Thus the price vector p is a WE
price vector.
Figure 3.1-7: PE allocation
Essential Microeconomics -31-
© John Riley
Define the transfer payment ˆ( ),h h hT p x hω= ⋅ − ∈H .
Because ˆh h
h h
x ω∈ ∈
=∑ ∑H H
the sum of these transfers is zero so this is a feasible redistribution of wealth.
The budget constraint ˆh hp x p x⋅ ≤ ⋅ can be rewritten as follows: h h hp x p Tω⋅ ≤ ⋅ + .
Then given transfers ,hT h∈H , the price vector p is a WE price vector.
Essential Microeconomics -32-
© John Riley
Proposition 3.1-3: Second welfare theorem for an exchange economy
In an exchange economy with endowments ,{ }hhω ∈H , suppose that ( )hU x , is continuously
differentiable, quasi concave on n+and that ( ) 0
hh
h
U xx
∂>>
∂ , h∈H . Then any PE allocation ˆ{ }h
hx ∈H
where ˆ 0,hx h≠ ∈H , can be supported by a price vector 0p > .
For expositional simplicity, consider a two person economy. The generalization is direct.
The idea on the proof is to argue that a PE allocation must be the solution to a maximization problem
and then show that the associated shadow prices are no-trade WE prices.
*
Essential Microeconomics -33-
© John Riley
Proposition 3.1-3: Second welfare theorem for an exchange economy
In an exchange economy with endowments ,{ }hhω ∈H , suppose that ( )hU x , is continuously
differentiable, quasi concave on n+and that ( ) 0
hh
h
U xx
∂>>
∂ , h∈H . Then any PE allocation ˆ{ }h
hx ∈H
where ˆ 0,hx h≠ ∈H , can be supported by a price vector 0p > .
For expositional simplicity, consider a two person economy. The generalization is direct.
The idea on the proof is to argue that a PE allocation must be the solution to a maximization problem
and then show that the associated shadow prices are no-trade WE prices.
Proof: If ˆ ˆ,A Bx x is a PE allocation then
,
ˆ ˆarg { ( ) | , ( ) ( )}A B
A A A A B A B B B B B
x xx Max U x x x U x U xω ω= + ≤ + ≥ . (3.1-2)
Class exercise: Explain why the assumptions imply that the Kuhn-Tucker conditions are necessary
conditions.
Essential Microeconomics -34-
© John Riley
The Lagrangian for the optimization problem (3.1-2) is
ˆ( ) ( ) ( ( ) ( ))A A A B A B B B B BU x x x U x U xν ω ω μ= + + − − + −L .
Kuhn-Tucker conditions.
ˆ( ) 0A
AA A
U xx x
ν∂ ∂= − ≤
∂ ∂L , where ˆ ˆ( ( ) ) 0
AA A
A
Ux xx
ν∂⋅ − =∂
. (3.1-3)
ˆ( ) 0B
BB A
U xx x
μ ν∂ ∂= − ≤
∂ ∂L , where ˆ ˆ( ( ) ) 0
BB B
B
Ux xx
μ ν∂⋅ − =
∂. (3.1-4)
ˆ ˆ 0A B A Bx xω ων∂
= + − − ≥∂L , where ˆ ˆ( ) 0A B A Bx xν ω ω⋅ + − − = . (3.1-5)
***
Essential Microeconomics -35-
© John Riley
The Lagrangian for the optimization problem (3.1-2) is
ˆ( ) ( ) ( ( ) ( ))A A A B A B B B B BU x x x U x U xν ω ω μ= + + − − + −L .
Kuhn-Tucker conditions.
ˆ( ) 0A
AA A
U xx x
ν∂ ∂= − ≤
∂ ∂L , where ˆ ˆ( ( ) ) 0
AA A
A
Ux xx
ν∂⋅ − =∂
. (3.1-3)
ˆ( ) 0B
BB A
U xx x
μ ν∂ ∂= − ≤
∂ ∂L , where ˆ ˆ( ( ) ) 0
BB B
B
Ux xx
μ ν∂⋅ − =
∂. (3.1-4)
ˆ ˆ 0A B A Bx xω ων∂
= + − − ≥∂L , where ˆ ˆ( ) 0A B A Bx xν ω ω⋅ + − − = . (3.1-5)
Because 0A
A
Ux
∂>>
∂ it follows from (3.1-3) that 0ν >> . From (3.1-5) it then follows that
ˆ ˆ 0A B A Bx xω ω+ − − = . (3.1-6)
**
Essential Microeconomics -36-
© John Riley
The Lagrangian for the optimization problem (3.1-2) is
ˆ( ) ( ) ( ( ) ( ))A A A B A B B B B BU x x x U x U xν ω ω μ= + + − − + −L .
Kuhn-Tucker conditions.
ˆ( ) 0A
AA A
U xx x
ν∂ ∂= − ≤
∂ ∂L , where ˆ ˆ( ( ) ) 0
AA A
A
Ux xx
ν∂⋅ − =∂
. (3.1-3)
ˆ( ) 0B
BB A
U xx x
μ ν∂ ∂= − ≤
∂ ∂L , where ˆ ˆ( ( ) ) 0
BB B
B
Ux xx
μ ν∂⋅ − =
∂. (3.1-4)
ˆ ˆ 0A B A Bx xω ων∂
= + − − ≥∂L , where ˆ ˆ( ) 0A B A Bx xν ω ω⋅ + − − = . (3.1-5)
Because 0A
A
Ux
∂>>
∂ it follows from (3.1-3) that 0ν >> . From (3.1-5) it then follows that
ˆ ˆ 0A B A Bx xω ω+ − − = . (3.1-6)
Because ˆ 0Bx > and 0B
B
Ux
∂>>
∂ it follows from (3.1-4) that 0μ > .
*
Essential Microeconomics -37-
© John Riley
The Lagrangian for the optimization problem (3.1-2) is
ˆ( ) ( ) ( ( ) ( ))A A A B A B B B B BU x x x U x U xν ω ω μ= + + − − + −L .
Kuhn-Tucker conditions.
ˆ( ) 0A
AA A
U xx x
ν∂ ∂= − ≤
∂ ∂L , where ˆ ˆ( ( ) ) 0
AA A
A
Ux xx
ν∂⋅ − =∂
. (3.1-3)
ˆ( ) 0B
BB A
U xx x
μ ν∂ ∂= − ≤
∂ ∂L , where ˆ ˆ( ( ) ) 0
BB B
B
Ux xx
μ ν∂⋅ − =
∂. (3.1-4)
ˆ ˆ 0A B A Bx xω ων∂
= + − − ≥∂L , where ˆ ˆ( ) 0A B A Bx xν ω ω⋅ + − − = . (3.1-5)
Because 0A
A
Ux
∂>>
∂ it follows from (3.1-3) that 0ν >> . From (3.1-5) it then follows that
ˆ ˆ 0A B A Bx xω ω+ − − = . (3.1-6)
Because ˆ 0Bx > and 0B
B
Ux
∂>>
∂ it follows from (3.1-4) that 0μ > .
Now consider an economy with endowments ˆ ˆh hxω = , h∈H and consider the price vector p ν= .
Essential Microeconomics -38-
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Consumer h chooses ˆarg { ( ) | }h
h h h h h
xx Max U x x xν ν= ⋅ ≤ ⋅ .
The FOC for this optimization problem are
( ) 0h
h hh h
U xx x
λ ν∂ ∂= − ≤
∂ ∂L , where ( ( ) ) 0
hh h h
h
Ux xx
λ ν∂− =
∂.
Moreover, because ( )hU ⋅ is quasi-concave the FOC is also sufficient. Choose 1Aλ = and 1/Bλ μ= .
Then, appealing to (3.1-3) and (3.1-4), the FOC hold at ˆ ,h hx x h= ∈H .
**
Essential Microeconomics -39-
© John Riley
Consumer h chooses ˆarg { ( ) | }h
h h h h h
xx Max U x x xν ν= ⋅ ≤ ⋅ .
The FOC for this optimization problem are
( ) 0h
h hh h
U xx x
λ ν∂ ∂= − ≤
∂ ∂L , where ( ( ) ) 0
hh h h
h
Ux xx
λ ν∂− =
∂.
Moreover, because ( )hU ⋅ is quasi-concave the FOC is also sufficient. Choose 1Aλ = and 1/Bλ μ= .
Then, appealing to (3.1-3) and (3.1-4), the FOC hold at ˆ ,h hx x h= ∈H .
Thus at the price p ν= no consumer wishes to trade. Therefore supply equals demand and so the price
vector is a WE price vector.
*
Essential Microeconomics -40-
© John Riley
Consumer h chooses ˆarg { ( ) | }h
h h h h h
xx Max U x x xν ν= ⋅ ≤ ⋅ .
The FOC for this optimization problem are
( ) 0h
h hh h
U xx x
λ ν∂ ∂= − ≤
∂ ∂L , where ( ( ) ) 0
hh h h
h
Ux xx
λ ν∂− =
∂.
Moreover, because ( )hU ⋅ is quasi-concave the FOC is also sufficient. Choose 1Aλ = and 1/Bλ μ= .
Then, appealing to (3.1-3) and (3.1-4), the FOC hold at ˆ ,h hx x h= ∈H .
Thus at the price p ν= no consumer wishes to trade. Therefore supply equals demand and so the price
vector is a WE price vector.
Finally define transfers ˆ( )h h hT xν ω= ⋅ − . Appealing to (3.1-2), the sum of these transfers is zero.
Consumer h’s budget constraint with these transfers is
ˆh h h hx T xν ν ω ν⋅ ≤ ⋅ + = ⋅ .
Thus the PE allocation is achievable as a WE with the appropriate transfer payments among
consumers.
Q.E.D.
Essential Microeconomics -41-
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Homothetic Preferences
Suppose that the two individuals in the economy (Alex and Bev) have different convex and
homothetic preferences. At the aggregate endowment, 1 2( , ),ω ω Alex has a stronger preference for
commodity 1 than Bev. That is, Alex is willing to give up more units of commodity 2 than Bev in
exchange for an additional unit of commodity 1.
Assumption: Differing Intensity of preferences
At the aggregate endowment, Alex has a stronger preference for commodity 1 than Bev.
1 2 1 21 2 1 2
( , ) ( , )/ /A A B B
A BU U U UMRS MRSx x x x
∂ ∂ ∂ ∂ω ω ω ω∂ ∂ ∂ ∂
= > = (3.1-7)
This is depicted in Figure 3.1-9.
Figure 3.1-9: Alex has a stronger preference for commodity 1
Essential Microeconomics -42-
© John Riley
We now explore the implications of this
assumption for the PE allocations.
Consider the PE allocation C in the
interior of the Edgeworth box.
Class exercises
1. Explain why all PE allocation lie below
the diagonal.
2. Explain why the allocations in the yellow and
dark blue regions are not PE.
Thus any other PE allocation C′preferred by Alex must lie above the line AO D . Because Alex’s MRS
is constant along this line, the marginal rate of substitution at C′ will be higher, reflecting the greater
influence of Alex’s stronger preference for commodity 1. Then
2 2
1 1
,h h
h h
x x hx xC C
< ∈′
H , and ( ) ( ),h hMRS C MRS C h′ > ∈H .
F
D
Commodity 1
Commodity 2
Fig 3.1‐10: Pareto efficient allocations
Essential Microeconomics -43-
© John Riley
We summarize these results below.
Proposition 3.1-4: Pareto Efficient Allocations With Homothetic Preferences
In the 2 2× exchange economy, suppose each consumer has homothetic preferences. Suppose also that
at the aggregate endowment, consumer A has a stronger preference for commodity 1. Then at any
interior efficient allocation,
2 2
1 1
A B
A B
x xx x
< .
Moreover, along the locus of efficient allocations, as consumer A’s utility rises, the consumption ratio
2 1/h hx x and marginal rate of substitution of 1x for 2x of both consumers rises.
Note that if Alex become relatively more wealthy so that the WE moves from C to C′ , the equilibrium
MRS rises. Thus 1 2/p p , the equilibrium relative price of commodity 1 rises.
Intuitively, since Alex has a stronger preference for commodity 1, the higher endowment, the more the
relative price reflects his preferences.
Essential Microeconomics -44-
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A closer look at the second welfare theorem
The economy
Commodities are private: Consumer {1,..., }h H∈ =H has preferences over his own consumption
vector 1( ,...,. )h h hnx x x=
Consumption set: Preferences are defined over the convex set h nX ⊂ .
Endowments: Consumer h has an endowment vector h hXω ∈ .
Consumption allocation: { }hhx ∈H where ,h hx X h∈ ∈H .
Aggregate consumption: h
h
x x∈
= ∑H
.
Aggregate endowment is h
h
ω ω∈
= ∑H
.
Excess demand: z x ω= − z
Essential Microeconomics -45-
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Feasible Allocation:
An allocation { }hhx ∈H satisfying 0z x ω= − ≤ .
***
Essential Microeconomics -46-
© John Riley
Feasible Allocation:
An allocation { }hhx ∈H satisfying 0z x ω= − ≤ .
Pareto-Efficient Allocation
A feasible allocation ˆ{ }hhx ∈H , is Pareto-efficient if there is no other feasible plan that is strictly
preferred by at least one consumer and weakly preferred by all consumers.
**
Essential Microeconomics -47-
© John Riley
Feasible Allocation:
An allocation { }hhx ∈H satisfying 0z x ω= − ≤ .
Pareto-Efficient Allocation
A feasible allocation ˆ{ }hhx ∈H , is Pareto-efficient if there is no other feasible plan that is strictly
preferred by at least one consumer and weakly preferred by all consumers.
Price-Taking
Let 0p ≥ be the price vector. Consumers are price takers. Consumer h has an endowment hω .
She chooses a consumption bundle hx in her budget set { | }h h h hx X p x p ω∈ ⋅ ≤ ⋅ .
*
Essential Microeconomics -48-
© John Riley
Feasible Allocation:
An allocation { }hhx ∈H satisfying 0z x ω= − ≤ .
Pareto-Efficient Allocation
A feasible allocation ˆ{ }hhx ∈H , is Pareto-efficient if there is no other feasible plan that is strictly
preferred by at least one consumer and weakly preferred by all consumers.
Price-Taking
Let 0p ≥ be the price vector. Consumers are price takers. Consumer h has an endowment hω .
She chooses a consumption bundle hx in her budget set { | }h h h hx X p x p ω∈ ⋅ ≤ ⋅ .
Walrasian Equilibrium
Each consumer chooses the most preferred consumption plan in her budget set. That is,
( ) ( ), for all such thath h h h h h hU x U x x p x p ω≥ ⋅ ≤ ⋅
Let hx x=∑ be the total consumption of the consumers. Excess demand is then z x ω= − .
Definition: Walrasian equilibrium prices
The price vector 0p ≥ is a Walrasian equilibrium price vector if there is no market in excess
demand ( 0)z ≤ and 0jp = for any market in excess supply ( 0)jz < .
Essential Microeconomics -49-
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Second welfare theorem
The earlier proof appealed to the Kuhn-Tucker conditions. As we have seen, these follow from the
Supporting Hyperplane Theorem.
We now dispense with differentiability assumption and appeal directly to the Supporting Hyperplane
Theorem.
If ˆ{ }hhx ∈H is PE it ˆ{ }h
hx ∈H must solve the following optimization problem.
1 1
{ }ˆ{ ( ) | ( ) ( ), 2,..., , ( ) 0, }
hh
h h h h h h h n
x h
Max U x U x U x h H x xω∈
+∈
≥ = − ≥ ∈∑H H
Consider the optimization problem when the aggregate supply is x.
( )PE x : 1 1
{ }ˆ{ ( ) | ( ) ( ), 2,..., , ( ) 0, }
hh
h h h h h h h n
x h
Max U x U x U x h H x xω∈
+∈
≥ = − ≥ ∈∑H H
Define
1
1 1 1
{ } 1
ˆ( ) { ( ) | ( ) ( ), 2,..., , 0}h H
h
Hh h h h h
x h
V x Max U x U x U x h H x x= =
= ≥ = − ≥∑
(3.2-1)
Note that ˆ{ }hhx ∈H solves the optimization problem ( )PE ω .
Essential Microeconomics -50-
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Second welfare theorem
The earlier proof appealed to the Kuhn-Tucker conditions. As w have seen, these follow from the
Supporting Hyperplane Theorem.
We now dispense with differentiability assumption and appeal directly to the Supporting Hyperplane
Theorem.
If ˆ{ }hhx ∈H is PE it ˆ{ }h
hx ∈H must solve the following optimization problem.
1 1
{ }ˆ{ ( ) | ( ) ( ), 2,..., , ( ) 0, }
hh
h h h h h h h n
x h
Max U x U x U x h H x xω∈
+∈
≥ = − ≥ ∈∑H H
*
Essential Microeconomics -51-
© John Riley
Lemma 3.2-1: Quasi-concavity of 1( )V ⋅
If ,hU h∈H is quasi-concave then so is the indirect utility function 1( )V ⋅ .
Proof: Class exercise.
Proposition 3.2-2: Second Welfare Theorem for an Exchange Economy
Consumer h∈H has an endowment h nω +∈ . The consumption set for each individual hX is the
positive orthant n+ . Suppose also that utility functions ( ),hU h⋅ ∈H are continuous, quasi-concave
and strictly increasing. If ˆ{ }h
hx ∈H is PE such that ˆ 0hx ≠ , h∈H then there exists a price vector 0p >
such that
ˆ ˆ( ) ( )h h h h h hU x U x p x p x> ⇒ ⋅ > ⋅ , h∈H
Essential Microeconomics -52-
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Proof: Appealing to the Lemma, 1( )V x is quasi-concave.
Also 1( )V ⋅ is strictly increasing because 1( )U ⋅ is strictly
increasing and any increment in the aggregate supply
can be allocated to the first consumer.
**
Fig 3.2-1: Supporting hyperplane
Essential Microeconomics -53-
© John Riley
Proof: Appealing to the Lemma, 1( )V x is quasi-concave.
Also 1( )V ⋅ is strictly increasing because 1( )U ⋅ is strictly
increasing and any increment in the aggregate supply
can be allocated to the first consumer.
An indifference curve for 1( )V ⋅ is depicted.
As we have noted that ˆ{ }hhx ∈H solves ( )PE x
if x ω= .
Moreover, because 1( )U ⋅ is strictly increasing,
1
ˆH
h
h
x ω=
=∑ . (3.2-2)
*
Fig 3.2-1: Supporting hyperplane
Essential Microeconomics -54-
© John Riley
Proof: Appealing to the Lemma, 1( )V x is quasi-concave.
Also 1( )V ⋅ is strictly increasing because 1( )U ⋅ is strictly
increasing and any increment in the aggregate supply
can be allocated to the first consumer.
An indifference curve for 1( )V ⋅ is depicted.
As we have noted that ˆ{ }hhx ∈H solves ( )PE x
if x ω= .
Moreover, because 1( )U ⋅ is strictly increasing,
1
ˆH
h
h
x ω=
=∑ . (3.2-2)
Because ω is on the boundary of the upper contour set 1 1{ | ( ) ( )}x V x V ω≥ , it follows from the
Supporting Hyperplane Theorem that there is a vector 0p ≠ , such that all the points in the upper
contour set lie in the set { | }x p x p ω⋅ ≥ ⋅
Fig 3.2-1: Supporting hyperplane
Essential Microeconomics -55-
© John Riley
Formally,
1 1 1 1( ) ( ) and ( ) ( )V x V p x p V x V p x pω ω ω ω> ⇒ ⋅ > ⋅ ≥ ⇒ ⋅ ≥ ⋅ . (3.2-3)
We now argue that the vector p must be positive.
If not, define 1( ,..., ) 0nδ δ δ= > such that
0jδ > if and only if 0jp < .
Then 1 1( ) ( )V Vω δ ω+ > and ( )p pω δ ω⋅ + < ⋅ .
But this contradicts (3.2-3) so p must be positive after all.
Fig 3.2-1: Supporting hyperplane
Essential Microeconomics -56-
© John Riley
To complete the proof we appeal to (3.2-1) - (3.2-3).
1
1 1 1
{ } 1
ˆ( ) { ( ) | ( ) ( ), 2,..., , 0}h H
h
Hh h h h h
x hV x Max U x U x U x h H x x
= =
= ≥ = − ≥∑
(3.2-1)
1
ˆH
h
hx ω
=
=∑ (3.2-2)
1 1 1 1( ) ( ) and ( ) ( )V x V p x p V x V p x pω ω ω ω> ⇒ ⋅ > ⋅ ≥ ⇒ ⋅ ≥ ⋅ . (3.2-3)
From (3.2-3)
1
ˆ( ) ( ), 1,...,H
h h h h h
hU x U x h H p x p x p ω
=
≥ = ⇒ ⋅ = ⋅ ≥ ⋅∑ . (3.2-4)
*
Essential Microeconomics -57-
© John Riley
To complete the proof we appeal to (3.2-1) - (3.2-3).
1
1 1 1
{ } 1
ˆ( ) { ( ) | ( ) ( ), 2,..., , 0}h H
h
Hh h h h h
x hV x Max U x U x U x h H x x
= =
= ≥ = − ≥∑
(3.2-1)
1
ˆH
h
hx ω
=
=∑ (3.2-2)
1 1 1 1( ) ( ) and ( ) ( )V x V p x p V x V p x pω ω ω ω> ⇒ ⋅ > ⋅ ≥ ⇒ ⋅ ≥ ⋅ . (3.2-3)
From (3.2-3)
1
ˆ( ) ( ), 1,...,H
h h h h h
hU x U x h H p x p x p ω
=
≥ = ⇒ ⋅ = ⋅ ≥ ⋅∑ . (3.2-5)
Substituting for ω from (3.2-2) it follows that
1 1
ˆ ˆ( ) ( ), 1,...,H H
h h h h h h
h hU x U x h H p x p x
= =
≥ = ⇒ ⋅ ≥ ⋅∑ ∑ .
Setting ˆ ,k kx x k h= ≠ , we may then conclude that for consumer h,
ˆ ˆ( ) ( )h h h h h hU x U x p x p x≥ ⇒ ⋅ ≥ ⋅ .
Essential Microeconomics -58-
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It remains to show that any strictly preferred bundle costs strictly more.
Suppose instead that ˆ ˆ( ) ( ) andh h h h h hU x U x p x p x> ⋅ = ⋅ .
Then for all (0,1)λ∈ , ˆh hp x p xλ⋅ < ⋅ .
Also because ( )hU ⋅ is continuous, for all λ sufficiently close to 1,
ˆ( ) ( )h h h hU x U xλ > .
But this cannot be true since we have just shown that
ˆ ˆ( ) ( )h h h h h hU x U x p x p x≥ ⇒ ⋅ ≥ ⋅
Hence
ˆ ˆ( ) ( )h h h h h hU x U x p x p x> ⇒ ⋅ > ⋅ .
Q.E.D.