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EC487 Advanced Microeconomics, Part I:Lecture 4
Leonardo Felli
32L.LG.04
20 October, 2017
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Marshallian Demands as Correspondences
We consider now the case in which consumers’ preferences arestrictly monotonic and weakly convex. In this case Marshalliandemands may be correspondence. Assume they are.
Definition
A correspondence is defined as a mapping F : X ⇒ Y such thatF (x) ⊂ Y for all x ∈ X .
Definition
The graph of a correspondence F : X ⇒ Y is the set:
G (F ) = {(x , y) ∈ X × Y | y ∈ F (x)}
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Fixed Point Theorem
Definition
A fixed point for a correspondence F : X ⇒ Y is a vector x∗ suchthat
x∗ ∈ F (x∗)
Theorem (Kakutani’s Fixed Point Theorem)
Let X be a compact, convex and non-empty set in RN . LetF : X ⇒ X be a correspondence.
Assume that G (F ) is closed and that F (x) is convex for everyx ∈ X then F has a fixed point.
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Kakutani’s Fixed Point Theorem
Here is an alternative statement of Kakutani’s Fixed PointTheorem (equivalent).
Theorem (Kakutani’s Fixed Point Theorem)
Let X be a compact, convex and non-empty set in RN . LetF : X ⇒ X be a correspondence such that:
I F is non-empty;
I F is convex valued;
I F is upper-hemi-continuous.
Then there exists a vector x∗ ∈ X such that:
x∗ ∈ F (x∗)
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Upper-hemi-continuity
Definition
A correspondence F : X ⇒ Y where X and Y are compact, convexsubsets of Euclidean space is upper-hemi-continuous if and only ifgiven the two converging sequences {xn} ⊂ X and {yn} ⊂ Y suchthat:
{xn}∞n=1 → x ∈ X ; {yn}∞n=1 → y ∈ Y
and:yn ∈ F (xn) ∀n
it is the case thaty ∈ F (x).
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Upper-hemi-continuity (cont’d)
Notice that
Result
Upper-hemi-continuity of the correspondence F (·) is equivalent toF (·) having a closed graph only if Y is a compact set.
Notice that if we consider a degenerate correspondence y = F (x)upper-hemi-continuity of F (x) is equivalent to continuity of thisfunction and implies that its graph is closed.
However, a closed graph does not imply that the function iscontinuous: example of a discontinuous function with a closedgraph is:
F (x) =
{1x if x 6= 03 if x = 0
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Existence of Walrasian Equilibrium
Theorem (Existence Theorem of Walrasian Equilibrium)
Let the excess demand function Z (p) be such that:
1. Z (p) is upper-hemi-continuous;
2. Z (p) is convex-valued;
3. Z (p) is bounded;
4. Z (p) homogeneous of degree 0;
5. Z (p) satisfies Walras Law: p Z (p) = 0;
then there exist a vector of prices p∗ and an induced allocation x∗
such that:Z (p∗) ≤ 0.
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Existence of Walrasian Equilibrium (cont’d)
Proof: Once again we shall start from a normalization of prices:p ∈ ∆L.
In this way the excess demand function will be such that:
Z : ∆L ⇒ RL
We then consider the following correspondence:
p ⇒ Z (p)⇒ p′
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Existence of Walrasian Equilibrium (cont’d)
Where we define p′ so that p′ Z (p) is maximized.
In other words, denote Z a vector in RL and m(Z ) the followingset of price vectors p̂:
m(Z ) = arg maxp̂
p̂ Z
s.t. p̂ ∈ ∆L(1)
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Existence of Walrasian Equilibrium (cont’d)
Claim
The set m(Z ) is convex.
Proof: Consider p ∈ ∆L and p′ ∈ ∆L that solves (1).
Then necessarily: p Z = p′ Z and for every λ ∈ [0, 1]:
[λ p + (1− λ) p′]Z = p Z = p′ Z .
Therefore: [λ p + (1− λ) p′] ∈ m(Z )
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Existence of Walrasian Equilibrium (cont’d)
Claim
The set m(Z ) is upper-hemi-continuous.
In other words, consider the following two sequences:
{Zn} → Z ∗
and{pn} → p∗
such thatpn ∈ m(Zn) ∀n
thenp∗ ∈ m(Z ∗).
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Existence of Walrasian Equilibrium (cont’d)
Proof: Suppose this is not true then
p∗ 6∈ m(Z ∗)
in other words there exists p̄ 6= p∗ such that
p̄ Z ∗ > p∗ Z ∗ (2)
Since {Zn} → Z ∗ and {pn} → p∗ we get that:
p̄ Zn → p̄ Z ∗
andpn Zn → p∗ Z ∗
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Existence of Walrasian Equilibrium (cont’d)
Choose now n large enough or such that:
|p̄ Zn − p̄ Z ∗| < (ε/2)
|pn Zn − p∗ Z ∗| < (ε/2)(3)
Conditions (2) and (3) imply
p̄ Zn > p∗Z ∗ +ε
2and p∗Z ∗ +
ε
2> pnZn
so thatp̄ Zn > pn Zn
a contradiction of the assumption: pn ∈ m(Zn).
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Existence of Walrasian Equilibrium (cont’d)
Let g : ∆L ⇒ ∆L be defined as:
g(p) = m(Z (p)).
Result
The composition of two upper-hemi continuous and convex-valuedcorrespondences is itself upper-hemi-continuous and convex-valued.
Therefore if Z (p) and m(Z ) are upper-hemi-continuous andconvex-valued then g(p) = m(Z (p)) is upper-hemi-continuous andconvex-valued.
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Existence of Walrasian Equilibrium (cont’d)
By Kakutani’s Fixed Point Theorem there exists p∗ such that
p∗ ∈ g(p∗).
We still need to check that this price vector p∗ is indeed aWalrasian equilibrium price vector.
By definition of g(p) and the fact that p∗ ∈ g(p∗) we know that
p∗ ∈ arg maxp
p Z (p∗)
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Existence of Walrasian Equilibrium (cont’d)
In other words
p∗ Z (p∗) ≥ p Z (p∗) ∀p ∈ ∆L. (4)
By Walras Law we know that:
p∗ Z (p∗) = 0
From (4) we can then prove the following.
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Existence of Walrasian Equilibrium (cont’d)
Result
The price vector p∗ is such that: Z (p∗) ≤ 0
Proof: Assume by way of contradiction that there exists an ` ≤ Lsuch that Z`(p
∗) > 0.
Choose then p̂ = (0, . . . , 0, 1, 0, . . . , 0) where the digit 1 is in the`-th position.
We then obtain that p̂ ∈ ∆L and:
p̂ Z (p∗) > 0 = p∗ Z (p∗)
which contradicts (4).
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Existence of Walrasian Equilibrium (cont’d)
Therefore, we have proved that:
I there exists a vector of prices p∗ such that
Z (p∗) ≤ 0
I or that p∗ is a Walrasian equilibrium price vector.
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Properties of Walrasian Equilibrium: Definitions
Recall that x = {x1, . . . , x I} denotes an allocation.
Definition
An allocation x Pareto dominates an alternative allocation x̄ if andonly if:
ui (xi ) ≥ ui (x̄
i ) ∀i ∈ {1, . . . , I}
and for some i :
ui (xi ) > ui (x̄
i ).
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Properties of Walrasian Equilibrium: Definitions (cont’d)
In other words, the allocation x makes no one worse-off andsomeone strictly better-off.
Definition
An allocation x is feasible in a pure exchange economy if and onlyif:
I∑i=1
x i` ≤ ω̄` ∀` ∈ {1, . . . , L}.
Definition
An allocation x is Pareto efficient if and only if it is feasible andthere does not exist an other feasible allocation thatPareto-dominates x .
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Properties of Walrasian Equilibrium: Definitions (cont’d)
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Benevolent Central Planner
A standard way to identify a Pareto-efficient allocation is tointroduce a benevolent central planner that has the authority tore-allocate resources across consumers so as to exhaust anygains-from-trade available.
Result
An allocation x∗ is Pareto-efficient if and only if there exists avector of weights λ = (λ1, . . . , λI ), λi ≥ 0, for all i = 1, . . . , I andλh > 0 for at least one h ≤ I , such that x∗ solves the followingproblem:
maxx1,...,x I
I∑i=1
λi ui (xi )
s.tI∑
i=1
x i ≤ ω̄(5)
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Benevolent Central Planner (cont’d)
Proof: Only if: an allocation x∗ that solves problem (5) for avector of weights λ is Pareto efficient.
Assume by way of contradiction that the allocation x̂ that solves(5) is not Pareto efficient.
Then there exists a feasible allocation x̃ and at least an individual isuch that
ui (x̃i ) > ui (x̂
i ), uj(x̃j) ≥ uj(x̂
j) ∀j 6= i
Then, given (λ1, . . . , λI ), the allocation x̃ is feasible in problem (5)and achieves (or can be modified to achieve) a higher maximand.This contradicts the assumption that x̂ solves problem (5).
We come back to the if later on.
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First Fundamental Theorem of Welfare Economics
Theorem (First Welfare Theorem)
Consider a pure exchange economy such that:
I consumers’ preferences are weakly monotonic
I there exists a Walrasian equilibrium {p∗, x∗} of this economy
then the allocation x∗ is a Pareto-efficient allocation.
Proof: Assume that the theorem is not true.
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First Welfare Theorem: Proof
Contradiction hypothesis: There exists an allocation x such that
I∑i=1
x i ≤ ω̄
andui (x
i ) ≥ ui (xi ,∗) ∀i ≤ I
and for some i ≤ Iui (x
i ) > ui (xi ,∗)
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First Welfare Theorem: Proof (cont’d)
Result
Thenp∗x i ≥ p∗x i ,∗ ∀i ≤ I .
Proof: Assume that this is not true and there exists i ≤ I suchthat
p∗x i < p∗x i ,∗
Fromp∗x i ,∗ = p∗ωi
we then getp∗x i < p∗ωi
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First Welfare Theorem: Proof (cont’d)
This implies that there exists ε > 0 such that if we denote eT thevector eT = (1, . . . , 1)
p∗(x i + ε e
)< p∗ωi .
Monotonicity of preferences then implies that
ui (xi + ε e) > ui (x
i )
which together with the contradiction hypothesis gives:
ui (xi + ε e) > ui (x
i ,∗)
This contradicts x i ,∗ = x i (p∗).
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First Welfare Theorem: Proof (cont’d)
Result
By contradiction hypothesis for some i we have ui (xi ) > ui (x
i ,∗)then for the same i
p∗ x i > p∗x i ,∗.
Proof: Assume this is not the case.
Then it is possible to find a consumption bundle x i which isaffordable for i :
p∗x i ≤ p∗x i ,∗ = p∗ ωi
and yields a higher level of utility: ui (xi ) > ui (x
i ,∗).
This is a contradiction of the hypothesis x i ,∗ = x i (p∗).
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First Welfare Theorem: Proof (cont’d)
Adding up these conditions across consumers we obtain:
I∑i=1
p∗x i >I∑
i=1
p∗x i ,∗
orI∑
i=1
p∗x i >I∑
i=1
p∗x i ,∗ = p∗ω̄
that is
p∗
[I∑
i=1
x i − ω̄
]> 0
since we know that p∗` ≥ 0 then there exists an ` such that
I∑i=1
x i` > ω̄`
a contradiction of the feasibility of the allocation x .Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 29 / 45
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Implicit Assumptions
Notice that the hypotheses necessary for this Theorem to hold donot guarantee the existence of a Walrasian equilibrium.
Underlying assumptions are:
I perfectly competitive markets;
I every commodity has a corresponding market(no-externalities).
Consider now the converse question.
Suppose you have a pure exchange economy and you want theconsumer to achieve a given Pareto-efficient allocation.
Is there a way to achieve this allocation in a fully decentralized(hands-off) way?
Answer: redistribution of endowments.
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Second Fundamental Theorem of Welfare Economics
Theorem (Second Welfare Theorem)
Consider a pure exchange economy with consumers’ preferencessatisfying:
1. (weak) convexity;
2. continuity and strict monotonicity.
Let x∗ be a Pareto-efficient allocation such that x i ,∗` > 0 for every` ≤ L and every i ≤ I . Then there exists an endowmentre-allocation ω′ such that:
I∑i=1
ω′i =I∑
i=1
ωi
and for some p∗ the vector {p∗, x∗} is a Walrasian equilibriumgiven ω′.
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Separating Hyperplane Theorem
Theorem (Separating Hyperplane Theorem)
Let A and B be two disjoint and convex set in RN .Then there exists a vector p ∈ RN such that
p x ≥ p y
for every x ∈ A and every y ∈ B.
In other words there exists an hyperplane identified by the vector pthat separates the set A and the set B.
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Second Welfare Theorem: Proof
Proof: Consider
B i ={x i ∈ RL
+ | ui (x i ) > ui (xi ,∗)}
Notice that B i is convex since preferences are convex byassumption (utility function is quasi-concave).
Let
B =I∑
i=1
B i =
{x ∈ RL
+ | x =I∑
i=1
x i , x i ∈ B i
}
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Second Welfare Theorem: Proof (cont’d)
Result
The set B is convex.
Proof: Take x , x ′ ∈ B.
Now x ∈ B implies x =I∑
i=1
x i and x ′ ∈ B implies x ′ =I∑
i=1
x ′i .
Therefore
[λx + (1− λ)x ′] = λ
I∑i=1
x i + (1− λ)I∑
i=1
x ′i
=I∑
i=1
[λx i + (1− λ)x ′i ] ∈ B
since [λx i + (1− λ)x ′i ] ∈ B i and B i is convex.
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Second Welfare Theorem: Proof (cont’d)
Result
Let v =I∑
i=1
x i ,∗ then v 6∈ B
Proof: Assume that this is not the case: v ∈ B.
This means that there exist I consumption bundles x̂ i ∈ B i suchthat
v =I∑
i=1
x i ,∗ =I∑
i=1
x̂ i .
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Second Welfare Theorem: Proof (cont’d)
Now, Pareto-efficiency of x∗ implies that v is feasible:
v =I∑
i=1
x̂ i =I∑
i=1
ωi
and by definition of B i
ui (x̂i ) > ui (x
i ,∗)
for every i ≤ I .
This contradicts the Pareto-efficiency of x∗.
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Second Welfare Theorem: Proof (cont’d)
Result
There exists a p∗ such that:
p∗ x ≥ p∗ v = p∗I∑
i=1
x i ,∗ = p∗I∑
i=1
ωi ∀x ∈ B
Proof: It follows directly from the Separating HyperplaneTheorem. Indeed, the sets {v} and B satisfy the assumptions ofthe theorem.
We still need to show that the p∗ we have obtained is indeed aWalrasian equilibrium.
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Second Welfare Theorem: Proof (cont’d)
Result
Notice first p∗ ≥ 0.
Proof: Denote eTn = (0, . . . , 0, 1, 0, . . . , 0) where the digit 1 is inthe n-th position, n ≤ L.
Notice that strict monotonicity of preferences implies:
v + en ∈ B
therefore from the result above we have that:
p∗ (v + en) ≥ p∗ v
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Second Welfare Theorem: Proof (cont’d)
In other words:p∗ (v + en − v) ≥ 0
orp∗ en ≥ 0
which is equivalent to:p∗n ≥ 0
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Second Welfare Theorem: Proof (cont’d)
Result
For every consumer i ≤ I
ui (xi ) > ui (x
i ,∗)
impliesp∗ x i ≥ p∗x i ,∗
Proof: Let θ ∈ (0, 1).
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Second Welfare Theorem: Proof (cont’d)
Consider the allocation
w i = x i (1− θ)
and
wh = xh,∗ +x i θ
I − 1∀h 6= i
the allocation w is a redistribution of resources from i to everyh 6= i .
For a small enough θ by strict monotonicity we have that w isPareto-preferred to x∗.
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Second Welfare Theorem: Proof (cont’d)
Hence by the result above:
p∗I∑
i=1
w i ≥ p∗I∑
i=1
x i ,∗
or
p∗
x i (1− θ) +∑h 6=i
xh,∗ + x iθ
=
= p∗
x i +∑h 6=i
xh,∗
≥ p∗
x i ,∗ +∑h 6=i
xh,∗
which implies
p∗x i ≥ p∗x i ,∗.
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Second Welfare Theorem: Proof (cont’d)
Result
For every agent iui (x
i ) > ui (xi ,∗)
impliesp∗x i > p∗x i ,∗
Proof: The result above implies that p∗x i ≥ p∗x i ,∗
Therefore we just have to rule out p∗x i = p∗x i ,∗
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Second Welfare Theorem: Proof (cont’d)
Continuity and strict monotonicity of preferences imply that forsome scalar ξ ∈ (0, 1) close to 1 we have
ui (ξ xi ) > ui (x
i ,∗)
and therefore by the last result above
p∗ξ x i ≥ p∗x i ,∗ (6)
If now p∗x i = p∗x i ,∗ > 0 from p∗ > 0 and x i ,∗ > 0 it follows that
p∗ξ x i < p∗x i ,∗
which contradicts (6).
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Second Welfare Theorem: Proof (cont’d)
The last two results imply that whenever ui (xi ) > ui (x
i ,∗) thenp∗x i ≥ p∗x i ,∗ with a strict inequality for some i .
This implies that the consumption bundles x i ,∗ maximizesconsumer i ’s utility subject to budget constraint.
MoreoverI∑
i=1
p∗x i ,∗ =I∑
i=1
p∗ωi
Let now ω′i = x i ,∗. This concludes the proof.
Notice that the assumptions of the Second Welfare Theorem arethe same that guarantee the existence of a Walrasian equilibrium.
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