EC487 Advanced Microeconomics, Part I: Lecture 4 Leonardo Felli - - PowerPoint PPT Presentation

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 are strictly monotonic and weakly convex. In this case Marshallian demands may be correspondence. Assume they are.

Definition

A correspondence is defined as a mapping F : X ⇒ Y such that F(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∗ such that x∗ ∈ F(x∗)

Theorem (Kakutani’s Fixed Point Theorem)

Let X be a compact, convex and non-empty set in RN. Let F : X ⇒ X be a correspondence. Assume that G(F) is closed and that F(x) is convex for every x ∈ 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 Point Theorem (equivalent).

Theorem (Kakutani’s Fixed Point Theorem)

Let X be a compact, convex and non-empty set in RN. Let F : X ⇒ X be a correspondence such that:

◮ F is non-empty; ◮ F is convex valued; ◮ 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, convex subsets of Euclidean space is upper-hemi-continuous if and only if given the two converging sequences {xn} ⊂ X and {yn} ⊂ Y such that: {xn}∞

n=1 → x ∈ X;

{yn}∞

n=1 → y ∈ Y

and: yn ∈ F(xn) ∀n it is the case that y ∈ 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 to F(·) 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 this function and implies that its graph is closed. However, a closed graph does not imply that the function is continuous: example of a discontinuous function with a closed graph is: F(x) = 1

x

if x = 0 3 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 following set of price vectors ˆ p: m(Z) = arg max

ˆ p

ˆ p Z s.t. ˆ p ∈ ∆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: {Z n} → Z ∗ and {pn} → p∗ such that pn ∈ m(Z n) ∀n then p∗ ∈ 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∗ ∈ m(Z ∗) in other words there exists ¯ p = p∗ such that ¯ p Z ∗ > p∗ Z ∗ (2) Since {Z n} → Z ∗ and {pn} → p∗ we get that: ¯ p Z n → ¯ p Z ∗ and pn Z n → 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: |¯ p Z n − ¯ p Z ∗| < (ε/2) |pn Z n − p∗ Z ∗| < (ε/2) (3) Conditions (2) and (3) imply ¯ p Z n > p∗Z ∗ + ε 2 and p∗Z ∗ + ε 2 > pnZ n so that ¯ p Z n > pn Z n a contradiction of the assumption: pn ∈ m(Z n).

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-valued correspondences is itself upper-hemi-continuous and convex-valued. Therefore if Z(p) and m(Z) are upper-hemi-continuous and convex-valued then g(p) = m(Z(p)) is upper-hemi-continuous and convex-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 a Walrasian equilibrium price vector. By definition of g(p) and the fact that p∗ ∈ g(p∗) we know that p∗ ∈ arg max

p

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 ℓ ≤ L such that Zℓ(p∗) > 0. Choose then ˆ p = (0, . . . , 0, 1, 0, . . . , 0) where the digit 1 is in the ℓ-th position. We then obtain that ˆ p ∈ ∆L and: ˆ p 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:

◮ there exists a vector of prices p∗ such that

Z(p∗) ≤ 0

◮ 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, . . . , xI} denotes an allocation.

Definition

An allocation x Pareto dominates an alternative allocation ¯ x if and

• nly if:

ui(xi) ≥ ui(¯ xi) ∀i ∈ {1, . . . , I} and for some i: ui(xi) > ui(¯ xi).

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 and someone strictly better-off.

Definition

An allocation x is feasible in a pure exchange economy if and only if:

I

• i=1

xi

ℓ ≤ ¯

ωℓ ∀ℓ ∈ {1, . . . , L}.

Definition

An allocation x is Pareto efficient if and only if it is feasible and there does not exist an other feasible allocation that Pareto-dominates x.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 20 / 45

Properties of Walrasian Equilibrium: Definitions (cont’d)

✲ ✻

x1

1

x1

2

✛ ❄

x2

1

x2

2

q q

¯ x x

q

xp

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 to introduce a benevolent central planner that has the authority to re-allocate resources across consumers so as to exhaust any gains-from-trade available.

Result

An allocation x∗ is Pareto-efficient if and only if there exists a vector of weights λ = (λ1, . . . , λI), λi ≥ 0, for all i = 1, . . . , I and λh > 0 for at least one h ≤ I, such that x∗ solves the following problem: max

x1,...,xI I

• i=1

λi ui(xi) s.t

I

• i=1

xi ≤ ¯ ω (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 a vector of weights λ is Pareto efficient. Assume by way of contradiction that the allocation ˆ x that solves (5) is not Pareto efficient. Then there exists a feasible allocation ˜ x and at least an individual i such that ui(˜ xi) > ui(ˆ xi), uj(˜ xj) ≥ uj(ˆ xj) ∀j = i Then, given (λ1, . . . , λI), the allocation ˜ x is feasible in problem (5) and achieves (or can be modified to achieve) a higher maximand. This contradicts the assumption that ˆ x 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:

◮ consumers’ preferences are weakly monotonic ◮ 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

xi ≤ ¯ ω and ui(xi) ≥ ui(xi,∗) ∀i ≤ I and for some i ≤ I ui(xi) > ui(xi,∗)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 25 / 45

First Welfare Theorem: Proof (cont’d)

Result

Then p∗xi ≥ p∗xi,∗ ∀i ≤ I. Proof: Assume that this is not true and there exists i ≤ I such that p∗xi < p∗xi,∗ From p∗xi,∗ = p∗ωi we then get p∗xi < 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 the vector eT = (1, . . . , 1) p∗ xi + ε e

• < p∗ωi.

Monotonicity of preferences then implies that ui(xi + ε e) > ui(xi) which together with the contradiction hypothesis gives: ui(xi + ε e) > ui(xi,∗) This contradicts xi,∗ = xi(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(xi,∗) then for the same i p∗ xi > p∗xi,∗. Proof: Assume this is not the case. Then it is possible to find a consumption bundle xi which is affordable for i: p∗xi ≤ p∗xi,∗ = p∗ ωi and yields a higher level of utility: ui(xi) > ui(xi,∗). This is a contradiction of the hypothesis xi,∗ = xi(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∗xi >

I

• i=1

p∗xi,∗

• r

I

• i=1

p∗xi >

I

• i=1

p∗xi,∗ = p∗¯ ω that is p∗ I

• i=1

xi − ¯ ω

• > 0

since we know that p∗

ℓ ≥ 0 then there exists an ℓ such that I

• i=1

xi

ℓ > ¯

ωℓ 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 do not guarantee the existence of a Walrasian equilibrium. Underlying assumptions are:

◮ perfectly competitive markets; ◮ every commodity has a corresponding market

(no-externalities). Consider now the converse question. Suppose you have a pure exchange economy and you want the consumer 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’ preferences satisfying:

• 1. (weak) convexity;
• 2. continuity and strict monotonicity.

Let x∗ be a Pareto-efficient allocation such that xi,∗

ℓ

> 0 for every ℓ ≤ L and every i ≤ I. Then there exists an endowment re-allocation ω′ such that:

I

• i=1

ω′i =

I

• i=1

ωi and for some p∗ the vector {p∗, x∗} is a Walrasian equilibrium given 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 p that 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 Bi =

• xi ∈ RL

+ | ui(xi) > ui(xi,∗)

• Notice that Bi is convex since preferences are convex by

assumption (utility function is quasi-concave). Let B =

I

• i=1

Bi =

• x ∈ RL

+ | x = I

• i=1

xi, xi ∈ Bi

• 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

xi and x′ ∈ B implies x′ =

I

• i=1

x′i. Therefore [λx + (1 − λ)x′] = λ

I

• i=1

xi + (1 − λ)

I

• i=1

x′i =

I

• i=1

[λxi + (1 − λ)x′i] ∈ B since [λxi + (1 − λ)x′i] ∈ Bi and Bi 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

xi,∗ then v ∈ B Proof: Assume that this is not the case: v ∈ B. This means that there exist I consumption bundles ˆ xi ∈ Bi such that v =

I

• i=1

xi,∗ =

I

• i=1

ˆ xi.

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

ˆ xi =

I

• i=1

ωi and by definition of Bi ui(ˆ xi) > ui(xi,∗) 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

xi,∗ = p∗

I

• i=1

ωi ∀x ∈ B Proof: It follows directly from the Separating Hyperplane

• Theorem. Indeed, the sets {v} and B satisfy the assumptions of

the theorem. We still need to show that the p∗ we have obtained is indeed a Walrasian 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 eT

n = (0, . . . , 0, 1, 0, . . . , 0) where the digit 1 is in

the 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

• r

p∗ 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(xi,∗) implies p∗ xi ≥ p∗xi,∗ 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 wi = xi (1 − θ) and wh = xh,∗ + xi θ I − 1 ∀h = i the allocation w is a redistribution of resources from i to every h = i. For a small enough θ by strict monotonicity we have that w is Pareto-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

wi ≥ p∗

I

• i=1

xi,∗

• r

p∗  xi(1 − θ) +

• h=i

xh,∗ + xiθ   = = p∗  xi +

• h=i

xh,∗   ≥ p∗  xi,∗ +

• h=i

xh,∗   which implies p∗xi ≥ p∗xi,∗.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 42 / 45

Second Welfare Theorem: Proof (cont’d)

Result

For every agent i ui(xi) > ui(xi,∗) implies p∗xi > p∗xi,∗ Proof: The result above implies that p∗xi ≥ p∗xi,∗ Therefore we just have to rule out p∗xi = p∗xi,∗

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 for some scalar ξ ∈ (0, 1) close to 1 we have ui(ξ xi) > ui(xi,∗) and therefore by the last result above p∗ξ xi ≥ p∗xi,∗ (6) If now p∗xi = p∗xi,∗ > 0 from p∗ > 0 and xi,∗ > 0 it follows that p∗ξ xi < p∗xi,∗ 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(xi,∗) then p∗xi ≥ p∗xi,∗ with a strict inequality for some i. This implies that the consumption bundles xi,∗ maximizes consumer i’s utility subject to budget constraint. Moreover

I

• i=1

p∗xi,∗ =

I

• i=1

p∗ωi Let now ω′i = xi,∗. This concludes the proof. Notice that the assumptions of the Second Welfare Theorem are the same that guarantee the existence of a Walrasian equilibrium.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 45 / 45