EC487 Advanced Microeconomics, Part I:Lecture 4
Leonardo Felli
32L.LG.04
20 October, 2017
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)}
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 2 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 3 / 45
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∗)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 4 / 45
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 5 / 45
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 6 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 7 / 45
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′
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 8 / 45
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)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 9 / 45
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 )
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 10 / 45
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 ∗).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 11 / 45
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 ∗
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 12 / 45
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 13 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 14 / 45
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∗)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 15 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 16 / 45
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 17 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 18 / 45
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 ).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 19 / 45
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 .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 20 / 45
Properties of Walrasian Equilibrium: Definitions (cont’d)
-
6
x11
x12
�
?
x21
x22
x̄
xqxp
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 21 / 45
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)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 22 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 23 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 24 / 45
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 ,∗)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 25 / 45
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 26 / 45
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∗).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 27 / 45
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∗).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 28 / 45
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
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 30 / 45
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′.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 31 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 32 / 45
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
}
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 33 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 34 / 45
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 .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 35 / 45
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∗.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 36 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 37 / 45
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 38 / 45
Second Welfare Theorem: Proof (cont’d)
In other words:p∗ (v + en − v) ≥ 0
orp∗ en ≥ 0
which is equivalent to:p∗n ≥ 0
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 39 / 45
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 40 / 45
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∗.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 41 / 45
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 ,∗.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 42 / 45
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 ,∗
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 43 / 45
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 44 / 45
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 45 / 45