EC487 Advanced Microeconomics, Part I: Lecture 3 Leonardo Felli - - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 3

Leonardo Felli

32L.LG.04

13 October, 2017

Bordered Hessian

Recall that when considering the cost minimization problem in the case of a technology with only two inputs f (x1, x2) we stated that the SOC are:

• f1(x∗)

f2(x∗) f1(x∗) f11(x∗) f12(x∗) f2(x∗) f21(x∗) f22(x∗)

• > 0

Why is this the case? Consider the cost minimization problem: min

{x1,x2}

w1 x1 + w2x2 s.t. f (x1, x2) ≥ y

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 2 / 48

Bordered Hessian (cont’d)

In other words: max

{x1,x2}

− (w1 x1 + w2x2) s.t. f (x1, x2) ≥ y The lagragian function is then: L(λ, x1, x2) = − (w1 x1 + w2x2) − λ [f (x1, x2) − y] According to the Lagrange method the solution to the cost minimization problem coincides with the solution to: max

{λ,x1,x2}

L(λ, x1, x2)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 3 / 48

Bordered Hessian (cont’d)

Notice now that the hessian matrix of the lagrangian H(x), where x = (x1, x2) is the bordered hessian      

∂2L ∂λ2 ∂2L ∂λ∂x1 ∂2L ∂λ∂x2 ∂2L ∂x1∂λ ∂2L ∂x2

1

∂2L ∂x1∂x2 ∂2L ∂x2∂λ ∂2L ∂x2∂x1 ∂2L ∂x2

2

      =   −f1(x) −f2(x) −f1(x) −λf11(x) −λf12(x) −f2(x) −λf21(x) −λf22(x)   Therefore the local SOC of cost minimization require that in a neighborhood of x∗: |H(x∗)| < 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 4 / 48

Bordered Hessian (cont’d)

Notice now that since λ > 0 sign   

• −f1(x∗)

−f2(x∗) −f1(x∗) −λf11(x∗) −λf12(x∗) −f2(x∗) −λf21(x∗) −λf22(x∗)

• 

  = = −sign   

• f1(x∗)

f2(x∗) f1(x∗) f11(x∗) f12(x∗) f2(x∗) f21(x∗) f22(x∗)

• 

  From here our SOC:

• f1(x∗)

f2(x∗) f1(x∗) f11(x∗) f12(x∗) f2(x∗) f21(x∗) f22(x∗)

• > 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 5 / 48

Competitive Equilibrium

Consider the entire economy, in which three main activities occur: production, consumption and trade. We shall focus first on a pure exchange economy (two activities

• nly: consumption and trade).

Consumers are born with endowments of commodities. They can either consume the endowments or trade them. Consider I = 2 consumers and L = 2 commodities.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 6 / 48

Pure Exchange Economy

In such case the consumption feasible set for every consumer is X i ∈ R2

+ and consumer i’s endowment is:

ωi = ωi

1

ωi

2

• The total endowment of commodity ℓ available in the economy is:

¯ ωℓ = ω1

ℓ + ω2 ℓ > 0

∀ℓ ∈ {1, 2} An allocation in this economy is then a pair of vectors x such that x = (x1, x2) = x1

1

x1

2

• ,

x2

1

x2

2

• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part I 13 October, 2017 7 / 48

Pure Exchange Economy (cont’d)

An allocation is feasible if and only if x1

ℓ + x2 ℓ ≤ ¯

ωℓ ∀ℓ ∈ {1, 2} An allocation is non-wasteful if and only if x1

ℓ + x2 ℓ = ¯

ωℓ ∀ℓ ∈ {1, 2} This economy can be represented in an Edgeworth box. In the example below we assume ω2

1 = ω1 2 = 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 8 / 48

Edgeworth Box

✲ ✻

x2 x1

✛ ❄ q

u1(x1, x2) u2(x1, x2) 1 2 ω

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 9 / 48

Edgeworth Box (cont’d)

Notice that in such an environment the income of each consumer is the market value of the consumer endowment: mi = p ωi where however p is determined in equilibrium. The budget set of consumer i is then: Bi(p) =

• xi ∈ R2

+ | p xi ≤ p ωi

For a vector of equilibrium prices p the budget sets of both consumers are two complementary sets in the Edgeworth box (slope of the separating line − p1

p2 ).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 10 / 48

Edgeworth Box (cont’d)

✲ ✻

x2 x1

✛ ❄ q

u1(x1, x2) u2(x1, x2) 1 2 ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 11 / 48

Edgeworth Box (cont’d)

The preferences of the two consumers are represented by two maps

• f indifference curves.

For any given level of prices we can represent the offer curve of each consumer: the consumption bundle that represent the optimal choice for each consumer. The offer curve necessarily passes through the endowment point. Indeed the allocation ω = (ω1, ω2) = ω1

1

ω1

2

• ,

ω2

1

ω2

2

• is always affordable hence each consumer must choose an optimal

consumption bundle that makes him/her at least as well off as at ω.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 12 / 48

Edgeworth Box (cont’d)

✲ ✻

x2 x1

✛ ❄ q

u1(x1, x2) u2(x1, x2) 1 2 ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 13 / 48

Edgeworth Box (cont’d)

✲ ✻

x2 x1

✛ ❄ q

u1(x1, x2) u2(x1, x2) 1 2 ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 14 / 48

Edgeworth Box (cont’d)

Given the preferences of the two consumers the only candidate to be an equilibrium price vector (if it exists) is a unique price vector that defines a unique budget constraint in the Edgeworth box tangent to indifference curves of both consumers. However if the tangency occur at two distinct points on the budget constraint then there will exist excess supply in one good, say ℓ = 2 and excess demand in the other good, say ℓ = 1. The allocation represented by the two tangency point is then not feasible.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 15 / 48

Edgeworth Box (cont’d)

✲ ✻

x2 x1

✛ ❄ q

u1(x1, x2) u2(x1, x2) 1 2 ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 16 / 48

Competitive Equilibrium

We define a market equilibrium as a situation in which markets clear, the consumers fulfil their desired purchases and the allocation obtained is feasible.

Definition

A Walrasian (competitive) equilibrium for the Edgeworth box economy is a price vector p∗ and an allocation x∗ = (x1,∗, x2,∗) such that ui(xi,∗) ≥ ui(xi) ∀xi ∈ Bi(p∗) and x1,∗

ℓ

+ x2,∗

ℓ

= ¯ ωℓ ∀ℓ ∈ {1, 2} This corresponds to an intersection of the offer curves: a point where the indifference curves of the two consumers are tangent to the unique budget constraint: the equilibrium allocation.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 17 / 48

Competitive Equilibrium (cont’d)

✲ ✻

x2 x1

✛ ❄ q

u1(x1, x2) u2(x1, x2) 1 2

q E

ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 18 / 48

Walras Law

Result (Walras Law)

The price vector p∗ is identified up to a degree of freedom: only the relative price matters. Proof: If the preferences of both consumers are locally non-satiated then the budget constraint of both consumers will be binding: p∗xi,∗ = p∗ωi ∀i ∈ {1, 2} If we sum the two budget constraint across consumers we get: p∗ x1,∗ + x2,∗ = p∗¯ ω which exhibits a linear dependence among the vectors of the equilibrium allocation (from here the degree of freedom).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 19 / 48

Walras Law (cont’d)

The more common formulation of Walras Law is:

Result (Walras Law)

In an pure-exchange economy with L commodities, L markets, if (L − 1) markets clear than necessarily the L-th market clear. The result is purely driven by binding budget constraints.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 20 / 48

Properties of Competitive Equilibrium

◮ Two main problems with a Walrasian equilibrium: existence

and uniqueness.

◮ Uniqueness is in general not a property of Walrasian equilibria. ◮ A Walrasian equilibrium might not exists (non-convexity of

preferences, unbounded demand).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 21 / 48

General Pure Exchange Economy

A general pure exchange economy with I consumers and L commodities is characterized by the following elements:

◮ i’s endowment vectors:

ωi =    ωi

1

. . . ωi

L

   ;

◮ i’s (locally-non-satiated) preferences represented by a utility

function ui(·).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 22 / 48

General Pure Exchange Economy (cont’d)

Denote

◮ the total endowment of each commodity ℓ as

¯ ωℓ =

I

• i=1

ωi

ℓ

∀ℓ ∈ {1, . . . , L}

◮ consumer i’s excess demand vector for any given distribution

• f endowments ω = {ω1, . . . , ωI} is:

zi(p) =    xi

1(p) − ωi 1

. . . xi

L(p) − ωi L

  

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 23 / 48

General Pure Exchange Economy (cont’d)

◮ the vector of aggregate excess demands as

Z(p) =    Z1(p) = I

i=1 zi 1(p)

. . . ZL(p) = I

i=1 zi L(p)

   In this pure exchange economy we can define a Walrasian equilibrium by means of the vector of aggregate excess demands in the following manner.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 24 / 48

Walrasian Equilibrium of a Pure Exchange Economy

Definition (Walrasian Equilibrium)

The Walrasian Equilibrium of a pure exchange economy is defined by a vector of prices p∗ and an induced allocation x∗ = {x1,∗(p∗), . . . , xI,∗(p∗)} such that all markets clear: Z(p∗) = 0 where Zℓ(p∗) =

I

• i=1
• xi,∗

ℓ (p∗) − ωi ℓ

• for ℓ = 1, . . . , L.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 25 / 48

Walras Law

◮ These L equations are not all independent, the reason being

Walras Law: each consumer Marshallian demand xi,∗(p) will be such that the consumer’s budget constraint will be binding: p∗xi,∗(p∗) = p∗ωi

◮ If we sum these budget constraint across the consumers we

get:

I

• i=1

p∗xi,∗(p∗) =

I

• i=1

p∗ωi

Result (Walras Law)

In other words: p∗ Z(p∗) = 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 26 / 48

Walras Law (cont’d)

◮ As seen before this condition introduces a degree of freedom

in the equilibrium price determination.

◮ In other words when L − 1 markets clear the L-th market has

to clear as well.

◮ Only the relative equilibrium price is determined in a

Walrasian equilibrium.

◮ An old approach to general equilibrium analysis consisted in

counting equations and unknowns.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 27 / 48

Walrasian Equilibrium: Definition

The modern approach is the one introduced by Debreu (1959). It starts from an alternative definition of Walrasian equilibrium.

Definition

A Walrasian equilibrium is a vector of prices p∗ and an allocation

• f resources x∗ associated to p∗ such that:

Z(p∗) ≤ 0 This alternative definition clearly allows for equilibrium excess supply.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 28 / 48

Walrasian Equilibrium: Preliminary Results

Given the definition above we can prove the following Lemma.

Lemma

p∗

ℓ ≥ 0 for every ℓ ∈ {1, . . . , L}.

Proof: Assume by way of contradiction that there exists ℓ such that pℓ < 0. The utility maximization problem is then: maxx u(x) s.t.

• h=ℓ

ph xh ≤ m − pℓ xℓ

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 29 / 48

Walrasian Equilibrium: Preliminary Results (cont’d)

If xℓ > 0 then pℓ xℓ < 0 therefore by increasing xℓ we do not decrease the objective function u(x). We can then increase xh, h = ℓ also unboundedly and u(x) → +∞. A contradiction to the existence of a solution to the utility maximization problem.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 30 / 48

Walrasian Equilibrium: Preliminary Results (cont’d)

Lemma

Let {p∗, x∗} be a Walrasian equilibrium then:

• 1. if p∗

ℓ > 0 then Zℓ(p∗) = 0;

• 2. if Zℓ(p∗) < 0 then p∗

ℓ = 0.

Proof: Walras Law implies that p∗ Z(p∗) = 0.

• r

L

• ℓ=1

p∗

ℓ Zℓ(p∗) = 0.

By the lemma above p∗

ℓ ≥ 0 while the definition of Walrasian

equilibrium implies Zℓ(p∗) ≤ 0. From here the result.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 31 / 48

Fixed Point

Definition (Fixed Point)

Consider a mapping F : RL → RL, any x∗ such that x∗ = F(x∗) is a fixed point of the mapping F.

Theorem (Brouwer Fixed Point Theorem)

Let S be a compact and convex set, and F : S → S a continuous mapping from S into itself. Then F has at least one fixed point in S.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 32 / 48

Existence of General Equilibrium

Consider now a pure exchange economy without any externality. Let Z(p) be the vector of excess demands that satisfies the following assumptions:

Assumption

• 1. Z(p) is single valued (it is a function).
• 2. Z(p) is continuous.
• 3. Z(p) is bounded.
• 4. Z(p) is homogeneous of degree 0.
• 5. Walras Law: p Z(p) = 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 33 / 48

Existence of General Equilibrium (cont’d)

Theorem (Existence Theorem of Walrasian Equilibrium)

Under assumptions 1–5 there exists a Walrasian Equilibrium price vector p∗ and an allocation x∗ such that Z(p∗) ≤ 0. Proof: Normalize the set of prices (recall that Walras Law leaves a degree of freedom in solving for the WE price vector p∗). Consider the prices in the L dimensional Simplex: S =

• p | p ≥ 0,

L

• ℓ=1

pℓ = 1

• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part I 13 October, 2017 34 / 48

Existence of General Equilibrium (cont’d)

Notice that by definition S is compact and convex. We shall

◮ define a continuous mapping from the Simplex into itself. ◮ use Brower fixed point theorem to obtain a fixed point of such

mapping.

◮ show that such a fixed point is indeed a Walrasian Equilibrium

price vector.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 35 / 48

Existence of General Equilibrium (cont’d)

Let β > 0 and define tℓ(p) = max {0, pℓ + β Zℓ(p)} which we normalize to be in S: qℓ(p) = tℓ L

ℓ=1 tℓ

We show next that the mapping from p into q is continuous.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 36 / 48

Existence of General Equilibrium (cont’d)

Indeed

◮ the mapping from p to t(p) is continuous: ◮ pℓ + β Zℓ(p) is continuous in p by assumption 2; ◮ a constant function is clearly continuous; ◮ the maximum of two continuous functions is also continuous. ◮ the mapping from t to q(p) is continuous provided that L

• ℓ=1

tℓ = 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 37 / 48

Existence of General Equilibrium (cont’d)

Lemma

Given the definition of tℓ above

L

• ℓ=1

tℓ = 0. Proof Recall that tℓ(p) = max {0, pℓ + β Zℓ(p)} By construction tℓ ≥ 0 for every ℓ = 1, . . . , L. Therefore

L

• ℓ=1

tℓ = 0 if and only if tℓ = 0 for every ℓ = 1, . . . , L.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 38 / 48

Existence of General Equilibrium (cont’d)

Assume by way of contradiction

L

• ℓ=1

tℓ = 0. From the lemma above we know that pℓ ≥ 0 therefore

◮ for every ℓ such that pℓ = 0 for tℓ = 0 we need Zℓ(p) ≤ 0. ◮ for every ℓ such that pℓ > 0 for tℓ = 0 we need Zℓ(p) < 0.

However, the latter case contradicts Walras Law:

L

• ℓ=1

p∗

ℓ Zℓ(p∗) = 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 39 / 48

Existence of General Equilibrium (cont’d)

Indeed, denote A(p) = {ℓ ≤ L | pℓ = 0}, and B(p) = {ℓ ≤ L | pℓ > 0}, by Walras Law: 0 =

L

• ℓ=1

pℓZℓ(p) =

• ℓ∈A(p)

pℓZℓ(p) +

• ℓ∈B(p)

pℓZℓ(p)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 40 / 48

Existence of General Equilibrium (cont’d)

Since by definition of A(p)

• ℓ∈A(p)

pℓZℓ(p) = 0 Walras Law implies:

• ℓ∈B(p)

pℓZℓ(p) = 0. This is a contradiction of pℓ > 0 and Zℓ(p) < 0 for every ℓ ∈ B(p).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 41 / 48

Existence of General Equilibrium (cont’d)

Therefore the mapping from p into q is continuous and maps a compact and convex set in itself. Brower fixed point theorem applies which means that there exists a fixed point p∗ such that q(p∗) = p∗. We still need to show that such a point is a Walrasian Equilibrium price vector.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 42 / 48

Existence of General Equilibrium (cont’d)

Consider first ℓ ∈ A(p∗) then p∗

ℓ = 0 by definition of A(p∗).

Further, being p∗ a fixed point qℓ(p∗) = p∗

ℓ = 0

This implies by definition of qℓ(p∗) and boundedness of Z(p) that tℓ(p∗) = max {0, p∗

ℓ + β Zℓ(p∗)} = max {0, β Zℓ(p∗)} = 0

Therefore, from β > 0, we have Zℓ(p∗) ≤ 0 for every ℓ ∈ A(p∗).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 43 / 48

Existence of General Equilibrium (cont’d)

Consider now ℓ ∈ B(p∗) then p∗

ℓ > 0 by definition of B(p∗).

Therefore by definition of tℓ(p∗): qℓ(p∗) = p∗

ℓ = p∗ ℓ + βZℓ(p∗)

L

ℓ=1 tℓ(p∗)

multiplying both sides by Zℓ(p∗) we get: p∗

ℓ Zℓ(p∗) = p∗ ℓ Zℓ(p∗) + β[Zℓ(p∗)]2

L

ℓ=1 tℓ(p∗)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 44 / 48

Existence of General Equilibrium (cont’d)

which summed over ℓ ∈ B(p∗) gives:

• ℓ∈B(p∗)

p∗

ℓ Zℓ(p∗) =

=

• ℓ∈B(p∗) p∗

ℓ Zℓ(p∗) + β ℓ∈B(p∗)[Zℓ(p∗)]2

L

ℓ=1 tℓ(p∗)

. Since by Walras Law

• ℓ∈B(p∗)

p∗

ℓ Zℓ(p∗) = 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 45 / 48

Existence of General Equilibrium (cont’d)

Therefore β

ℓ∈B(p∗)[Zℓ(p∗)]2

L

ℓ=1 tℓ(p∗)

= 0 Using the lemma above and tℓ(p∗) = 0 for every ℓ ∈ A(p∗) we get

L

• ℓ=1

tℓ =

• ℓ∈A(p∗)

tℓ +

• ℓ∈B(p∗)

tℓ =

• ℓ∈B(p∗)

tℓ = 0 which together with β > 0 implies

• ℓ∈B(p∗)

[Zℓ(p∗)]2 = 0 which implies Zℓ(p∗) = 0 for every ℓ ∈ B(p∗).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 46 / 48

Existence of General Equilibrium

In other words, we have proved that under assumptions 1–5 there exists a Walrasian Equilibrium price vector p∗ and an induced allocation x∗(p∗) such that:

◮ for every ℓ ∈ A(p∗) — for every ℓ such that p∗ ℓ = 0 — we

have that Zℓ(p∗) ≤ 0

◮ while for every ℓ ∈ B(p∗) — for every ℓ such that p∗ ℓ > 0 —

we have that Zℓ(p∗) = 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 47 / 48

Existence of General Equilibrium: Comments

Notice that this result implies that

◮ for all commodities that have a strictly positive equilibrium

prize the market clears: excess demand is zero;

◮ there may exists excess supply Zℓ(p∗) < 0 only for

commodities that are free (whose equilibrium price is zero).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 48 / 48