ec487 advanced microeconomics, part i: lecture 3econ.lse.ac.uk/staff/lfelli/teach/ec487 slides...

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 thecase of a technology with only two inputs f (x1, x2) we stated thatthe SOC are: ∣∣∣∣∣∣

0 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 costminimization problem coincides with the solution to:

max{λ,x1,x2}

L(λ, x1, x2)



Notice now that the hessian matrix of the lagrangian H(x), wherex = (x1, x2) is the bordered hessian

∂2L∂λ2

∂2L∂λ∂x1

∂2L∂λ∂x2

∂2L∂x1∂λ

∂2L∂x21

∂2L∂x1∂x2

∂2L∂x2∂λ

∂2L∂x2∂x1

∂2L∂x22

=

0 −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 aneighborhood of x∗:

|H(x∗)| < 0



Notice now that since λ > 0

sign

∣∣∣∣∣∣

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

∣∣∣∣∣∣ =

= −sign

∣∣∣∣∣∣

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

∣∣∣∣∣∣

From here our SOC:∣∣∣∣∣∣0 f1(x∗) f2(x∗)

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

∣∣∣∣∣∣ > 0


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 activitiesonly: 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.


Pure Exchange Economy

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

+ and consumer i ’s endowment is:

ωi =

(ωi1

ωi2

)

The total endowment of commodity ` available in the economy is:

ω̄` = ω1` + ω2

` > 0 ∀` ∈ {1, 2}

An allocation in this economy is then a pair of vectors x such that

x = (x1, x2) =

((x11x12

),

(x21x22

))


Pure Exchange Economy (cont’d)

An allocation is feasible if and only if

x1` + x2` ≤ ω̄` ∀` ∈ {1, 2}

An allocation is non-wasteful if and only if

x1` + x2` = ω̄` ∀` ∈ {1, 2}

This economy can be represented in an Edgeworth box. In theexample below we assume ω2

1 = ω12 = 0


Edgeworth Box

-

6

x2

x1

�

?q

u1(x1, x2)

u2(x1, x2)

1

2

ω


Edgeworth Box (cont’d)

Notice that in such an environment the income of each consumeris 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:

B i (p) ={x i ∈ R2

+ | p x i ≤ p ωi}

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

p2).



-

6

x2

x1

�

?q

u1(x1, x2)

u2(x1, x2)

1

2

ω.....

..........

..........

..........

..........

..........

..........

..........

..........

..........

........



The preferences of the two consumers are represented by two mapsof indifference curves.

For any given level of prices we can represent the offer curve ofeach consumer: the consumption bundle that represent the optimalchoice for each consumer.

The offer curve necessarily passes through the endowment point.

Indeed the allocation

ω = (ω1, ω2) =

((ω11

ω12

),

(ω21

ω22

))is always affordable hence each consumer must choose an optimalconsumption bundle that makes him/her at least as well off as atω.



-

6

x2

x1

�

?q

u1(x1, x2)

u2(x1, x2)

1

2

ω.....

..........

..........

..........

..........

..........

..........

..........

..........

..........

........



Given the preferences of the two consumers the only candidate tobe an equilibrium price vector (if it exists) is a unique price vectorthat defines a unique budget constraint in the Edgeworth boxtangent to indifference curves of both consumers.

However if the tangency occur at two distinct points on the budgetconstraint then there will exist excess supply in one good, say` = 2 and excess demand in the other good, say ` = 1.

The allocation represented by the two tangency point is then notfeasible.



-

6

x2

x1

�

?q

u1(x1, x2)

u2(x1, x2)

1

2

ω.....

..........

..........

..........

..........

..........

..........

..........

..........

..........

........


Competitive Equilibrium

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

Definition

A Walrasian (competitive) equilibrium for the Edgeworth boxeconomy is a price vector p∗ and an allocation x∗ = (x1,∗, x2,∗)such that

ui (xi ,∗) ≥ ui (x

i ) ∀x i ∈ B i (p∗)

andx1,∗` + x2,∗` = ω̄` ∀` ∈ {1, 2}

This corresponds to an intersection of the offer curves: a pointwhere the indifference curves of the two consumers are tangent tothe unique budget constraint: the equilibrium allocation.


Competitive Equilibrium (cont’d)

-

6

x2

x1

�

?q

u1(x1, x2)

u2(x1, x2)

1

2

q E

ω.....

..........

..........

..........

..........

..........

..........

..........

..........

..........

........


Walras Law

Result (Walras Law)

The price vector p∗ is identified up to a degree of freedom: onlythe relative price matters.

Proof: If the preferences of both consumers are locallynon-satiated then the budget constraint of both consumers will bebinding:

p∗x i ,∗ = p∗ωi ∀i ∈ {1, 2}

If we sum the two budget constraint across consumers we get:

p∗(x1,∗ + x2,∗

)= p∗ω̄

which exhibits a linear dependence among the vectors of theequilibrium allocation (from here the degree of freedom).


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.


Properties of Competitive Equilibrium

I Two main problems with a Walrasian equilibrium: existenceand uniqueness.

I Uniqueness is in general not a property of Walrasian equilibria.

I A Walrasian equilibrium might not exists (non-convexity ofpreferences, unbounded demand).


General Pure Exchange Economy

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

I i ’s endowment vectors:

ωi =

ωi1...ωiL

;

I i ’s (locally-non-satiated) preferences represented by a utilityfunction

ui (·).


General Pure Exchange Economy (cont’d)

Denote

I the total endowment of each commodity ` as

ω̄` =I∑

i=1

ωi` ∀` ∈ {1, . . . , L}

I consumer i ’s excess demand vector for any given distributionof endowments ω = {ω1, . . . , ωI} is:

z i (p) =

x i1(p)− ωi1

...x iL(p)− ωi

L


General Pure Exchange Economy (cont’d)

I the vector of aggregate excess demands as

Z (p) =

Z1(p) =∑I

i=1 zi1(p)

...

ZL(p) =∑I

i=1 ziL(p)

In this pure exchange economy we can define a Walrasianequilibrium by means of the vector of aggregate excess demands inthe following manner.


Walrasian Equilibrium of a Pure Exchange Economy

Definition (Walrasian Equilibrium)

The Walrasian Equilibrium of a pure exchange economy is definedby a vector of prices p∗ and an induced allocation

x∗ = {x1,∗(p∗), . . . , x I ,∗(p∗)}

such that all markets clear:

Z (p∗) = 0

where Z`(p∗) =

I∑i=1

(x i ,∗` (p∗)− ωi

`

)for ` = 1, . . . , L.


Walras Law

I These L equations are not all independent, the reason beingWalras Law: each consumer Marshallian demand x i ,∗(p) willbe such that the consumer’s budget constraint will be binding:

p∗x i ,∗(p∗) = p∗ωi

I If we sum these budget constraint across the consumers weget:

I∑i=1

p∗x i ,∗(p∗) =I∑

i=1

p∗ωi

Result (Walras Law)

In other words:p∗ Z (p∗) = 0


Walras Law (cont’d)

I As seen before this condition introduces a degree of freedomin the equilibrium price determination.

I In other words when L− 1 markets clear the L-th market hasto clear as well.

I Only the relative equilibrium price is determined in aWalrasian equilibrium.

I An old approach to general equilibrium analysis consisted incounting equations and unknowns.


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 allocationof resources x∗ associated to p∗ such that:

Z (p∗) ≤ 0

This alternative definition clearly allows for equilibrium excesssupply.


Walrasian Equilibrium: Preliminary Results

Given the definition above we can prove the following Lemma.

Lemma

p∗` ≥ 0 for every ` ∈ {1, . . . , L}.

Proof: Assume by way of contradiction that there exists ` suchthat p` < 0. The utility maximization problem is then:

maxx u(x)

s.t.∑h 6=`

ph xh ≤ m − p` x`


Walrasian Equilibrium: Preliminary Results (cont’d)

If x` > 0 then p` x` < 0 therefore by increasing x` we do notdecrease the objective function u(x).

We can then increase xh, h 6= ` also unboundedly and u(x)→ +∞.

A contradiction to the existence of a solution to the utilitymaximization problem.


Walrasian Equilibrium: Preliminary Results (cont’d)

Lemma

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

1. if p∗` > 0 then Z`(p∗) = 0;

2. if Z`(p∗) < 0 then p∗` = 0.

Proof: Walras Law implies that

p∗ Z (p∗) = 0. orL∑`=1

p∗` Z`(p∗) = 0.

By the lemma above p∗` ≥ 0 while the definition of Walrasianequilibrium implies Z`(p

∗) ≤ 0. From here the result.


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 onefixed point in S .


Existence of General Equilibrium

Consider now a pure exchange economy without any externality.

Let Z (p) be the vector of excess demands that satisfies thefollowing 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.


Existence of General Equilibrium (cont’d)

Theorem (Existence Theorem of Walrasian Equilibrium)

Under assumptions 1–5 there exists a Walrasian Equilibrium pricevector p∗ and an allocation x∗ such that

Z (p∗) ≤ 0.

Proof: Normalize the set of prices (recall that Walras Law leaves adegree of freedom in solving for the WE price vector p∗).

Consider the prices in the L dimensional Simplex:

S =

{p | p ≥ 0,

L∑`=1

p` = 1

}



Notice that by definition S is compact and convex.

We shall

I define a continuous mapping from the Simplex into itself.

I use Brower fixed point theorem to obtain a fixed point of suchmapping.

I show that such a fixed point is indeed a Walrasian Equilibriumprice vector.



Let β > 0 and define

t`(p) = max {0, p` + β Z`(p)}

which we normalize to be in S :

q`(p) =t`∑L`=1 t`

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



Indeed

I the mapping from p to t(p) is continuous:

I p` + β Z`(p) is continuous in p by assumption 2;

I a constant function is clearly continuous;

I the maximum of two continuous functions is also continuous.

I the mapping from t to q(p) is continuous provided that

L∑`=1

t` 6= 0.



Lemma

Given the definition of t` above

L∑`=1

t` 6= 0.

Proof Recall that

t`(p) = max {0, p` + β Z`(p)}

By construction t` ≥ 0 for every ` = 1, . . . , L.

ThereforeL∑`=1

t` = 0 if and only if t` = 0 for every ` = 1, . . . , L.



Assume by way of contradictionL∑`=1

t` = 0.

From the lemma above we know that p` ≥ 0 therefore

I for every ` such that p` = 0 for t` = 0 we need Z`(p) ≤ 0.

I for every ` such that p` > 0 for t` = 0 we need Z`(p) < 0.

However, the latter case contradicts Walras Law:

L∑`=1

p∗` Z`(p∗) = 0.



Indeed, denoteA(p) = {` ≤ L | p` = 0},

andB(p) = {` ≤ L | p` > 0},

by Walras Law:

0 =L∑`=1

p`Z`(p) =∑`∈A(p)

p`Z`(p) +∑`∈B(p)

p`Z`(p)



Since by definition of A(p)∑`∈A(p)

p`Z`(p) = 0

Walras Law implies: ∑`∈B(p)

p`Z`(p) = 0.

This is a contradiction of p` > 0 and Z`(p) < 0 for every` ∈ B(p).



Therefore the mapping from p into q is continuous and maps acompact and convex set in itself.

Brower fixed point theorem applies which means that there exists afixed point p∗ such that q(p∗) = p∗.

We still need to show that such a point is a Walrasian Equilibriumprice vector.



Consider first ` ∈ A(p∗) then p∗` = 0 by definition of A(p∗).

Further, being p∗ a fixed point

q`(p∗) = p∗` = 0

This implies by definition of q`(p∗) and boundedness of Z (p) that

t`(p∗) = max {0, p∗` + β Z`(p

∗)} = max {0, β Z`(p∗)} = 0

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



Consider now ` ∈ B(p∗) then p∗` > 0 by definition of B(p∗).

Therefore by definition of t`(p∗):

q`(p∗) = p∗` =

p∗` + βZ`(p∗)∑L

`=1 t`(p∗)

multiplying both sides by Z`(p∗) we get:

p∗` Z`(p∗) =

p∗` Z`(p∗) + β[Z`(p

∗)]2∑L`=1 t`(p

∗)



which summed over ` ∈ B(p∗) gives:∑`∈B(p∗)

p∗` Z`(p∗) =

=

∑`∈B(p∗) p

∗` Z`(p

∗) + β∑

`∈B(p∗)[Z`(p∗)]2∑L

`=1 t`(p∗)

.

Since by Walras Law ∑`∈B(p∗)

p∗` Z`(p∗) = 0



Thereforeβ∑

`∈B(p∗)[Z`(p∗)]2∑L

`=1 t`(p∗)

= 0

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

L∑`=1

t` =∑

`∈A(p∗)

t` +∑

`∈B(p∗)

t` =∑

`∈B(p∗)

t` 6= 0

which together with β > 0 implies∑`∈B(p∗)

[Z`(p∗)]2 = 0

which implies Z`(p∗) = 0 for every ` ∈ B(p∗).


Existence of General Equilibrium

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

I for every ` ∈ A(p∗) — for every ` such that p∗` = 0 — wehave that

Z`(p∗) ≤ 0

I while for every ` ∈ B(p∗) — for every ` such that p∗` > 0 —we have that

Z`(p∗) = 0


Existence of General Equilibrium: Comments

Notice that this result implies that

I for all commodities that have a strictly positive equilibriumprize the market clears: excess demand is zero;

I there may exists excess supply Z`(p∗) < 0 only for

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


ec487 advanced microeconomics, part i: lecture 3econ.lse.ac.uk/staff/lfelli/teach/ec487 slides...

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