EC487 Advanced Microeconomics, Part I: Lecture 4econ.lse.ac.uk/staff/lfelli/teach/EC487 Slides Lecture 4.pdfEC487 Advanced Microeconomics, Part I: Lecture 4 Leonardo Felli 32L.LG.04

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.


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∗)


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).


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


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.


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′



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)



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 )



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 ∗).



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 ∗



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).



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.



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∗)



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.



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).



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.


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 ).


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 .


Properties of Walrasian Equilibrium: Definitions (cont’d)

-

6

x11

x12

�

?

x21

x22

qq

x̄

xqxp


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)


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.


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.


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 ,∗)


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



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∗).



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∗).



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.


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′.


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.


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

}


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.



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 .



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∗.



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.



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



In other words:p∗ (v + en − v) ≥ 0

orp∗ en ≥ 0

which is equivalent to:p∗n ≥ 0



Result

For every consumer i ≤ I

ui (xi ) > ui (x

i ,∗)

impliesp∗ x i ≥ p∗x i ,∗

Proof: Let θ ∈ (0, 1).



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∗.



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 ,∗.



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 ,∗



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).



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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EC487 Advanced Microeconomics, Part I: Lecture 4econ.lse.ac.uk/staff/lfelli/teach/EC487 Slides Lecture 4.pdfEC487 Advanced Microeconomics, Part I: Lecture 4 Leonardo Felli 32L.LG.04