Part1 P.Piacquadio€¦ · [email protected] September24,2014 P. Piacquadio ([email protected]) Micro 3200/4200 September 24, 2014 1 / 26. Outline 1 GeneralEquilibrium

Microeconomics 3200/4200:Part 1

P. Piacquadio

[email protected]

September 24, 2014

P. Piacquadio ([email protected]) Micro 3200/4200 September 24, 2014 1 / 26

Outline

1 General EquilibriumIntroductionAllocations, competitive allocations, and competitive equilibriaA 2 agents, 2 goods illustrationThe excess demand approachOn the equilibrium


A world with consumers and firms

We started with the analysis of a firm......then we looked at the problem of the consumer......we can now put them together with all other firms and consumerand understand how these interact.








The role of prices

We said that both the firm and the consumers are price takers: theyobserve a given price and behave accordingly.Now prices will play the crucial role:

I they will lead firms to produce the goods that are more desirable byconsumers;

I they will lead consumers to demand the goods that are cheaper toproduce.

In synthesis, prices will allow for supply and demand to meet exactlyin, what we shall call, a “general equilibrium.”


The role of prices






The role of prices






The ingredients (1)

Households:I Each household h = 1, ...,nhhas a utility function Uh;

Firms:I Each firm f = 1, ...,nf has a technology Φf ;

Resource stocks:I In the economy there are a certain amount of each resource i = 1, ...,n

available: Ri .


The ingredients (1)




available: Ri .


The ingredients (1)




available: Ri .


The ingredients (2)

Households:I xh

i is the quantity of good i consumed by household h;I xh =

(xh1 , ...,x

hi , ...,x

hn)is the consumption vector of household h;

I [x ] =[x1, ...,xh, ...,xnh

]is the vector of consumptions of all households;

I yh is the income of household h.

Firms:I qf

i is the net output (netput) of good i produced by firm f ;I qf =

(qf1 , ...,q

fi , ...,q

fn)is the net production vector of firm f ;

I [q] =[q1, ...,qf , ...,qnf

]is the vector of productions of all firms.


The ingredients (2)

Households:I xh

i is the quantity of good i consumed by household h;I xh =

(xh1 , ...,x

hi , ...,x

hn)is the consumption vector of household h;

I [x ] =[x1, ...,xh, ...,xnh

]is the vector of consumptions of all households;

I yh is the income of household h.

Firms:I qf

i is the net output (netput) of good i produced by firm f ;I qf =

(qf1 , ...,q

fi , ...,q

fn)is the net production vector of firm f ;

I [q] =[q1, ...,qf , ...,qnf

]is the vector of productions of all firms.


Outline



An allocation and a market allocation

An allocation defines a consumption vector for each household and anet output vector for each firm.

I It is denoted by a = ([x ] , [q]).

A market allocation defines a consumption vector for eachhousehold, a net output vector for each firm, and the prices availableon the market.

I It is denoted by a = ([x ] , [q] ,p).


An allocation and a market allocation

An allocation defines a consumption vector for each household and anet output vector for each firm.

I It is denoted by a = ([x ] , [q]).

A market allocation defines a consumption vector for eachhousehold, a net output vector for each firm, and the prices availableon the market.

I It is denoted by a = ([x ] , [q] ,p).


A competitive allocation

A competitive (market) allocation is a market allocationa = ([x ] , [q] ,p) such that:

I each households h maximizes its utility Uh at prices p;I each firm f maximizes its profits at prices p.


Reminder

xh solves the utility maximization problem of household h if xh is thesolution to:

maxxh

Uh(xh)

s.t.n

∑i=1

pixhi ≤ yh

qf solves the profit maximization problem of firm f if qf is thesolution to:

maxqf

n

∑i=1

piqfi s.t.Φf

(qf)≤ 0


Reminder

xh solves the utility maximization problem of household h if xh is thesolution to:

maxxh

Uh(xh)

s.t.n

∑i=1

pixhi ≤ yh

qf solves the profit maximization problem of firm f if qf is thesolution to:

maxqf

n

∑i=1

piqfi s.t.Φf

(qf)≤ 0


A competitive equilibrium

A competitive equilibrium (allocation) is a competitive allocationa = ([x ] , [q] ,p) in which the material balance condition is satisfied.

That is consumption is not larger than what is produced plus theavailable resources, or:

nh

∑h=1

xhi ≤

nf

∑f =1

qfi +

nh

∑h=1

Rhi .


A competitive equilibrium

A competitive equilibrium (allocation) is a competitive allocationa = ([x ] , [q] ,p) in which the material balance condition is satisfied.

That is consumption is not larger than what is produced plus theavailable resources, or:

nh

∑h=1

xhi ≤

nf

∑f =1

qfi +

nh

∑h=1

Rhi .


Household incomesThe income of a household h is given by the value of the resourcesavailable to h, i.e. Rh, plus that share of the profit of each firm f thatthe household owns.

Define ζ hf to be the share of (the profit Πf of) firm f in the hands of

household h.

Thus, the income of household h is:

yh =n

∑i=1

piRhi +

nf

∑f =1

ζhf Πf

Substituting Πf = ∑ni=1 piqf

i gives:

yh =n

∑i=1

pi

[Rh

i +nf

∑f =1

ζhf qf

i

].




household h.


yh =n

∑i=1

piRhi +

nf

∑f =1

ζhf Πf


i gives:

yh =n

∑i=1

pi

[Rh

i +nf

∑f =1

ζhf qf

i

].




household h.


yh =n

∑i=1

piRhi +

nf

∑f =1

ζhf Πf


i gives:

yh =n

∑i=1

pi

[Rh

i +nf

∑f =1

ζhf qf

i

].




household h.


yh =n

∑i=1

piRhi +

nf

∑f =1

ζhf Πf


i gives:

yh =n

∑i=1

pi

[Rh

i +nf

∑f =1

ζhf qf

i

].


Outline



An illustration

How to construct an “Edgeworth box”; the case of exchangeeconomy.

The contract curve.

Extending the “Edgeworth box” to production.


An illustration


The contract curve.



An illustration


The contract curve.



Outline



The excess demand function

Think of a hypothetical price for each good being announced on themarket.

The excess demand of good i at price p is the difference between thedemand of good i and the amount available (net output plusresources).

The excess demand function expresses such excess demand as afunction of prices:

Ei (p) := xi (p)︸︷︷︸demand for i

− qi (p)︸︷︷︸netput of i

− Ri︸︷︷︸resources of i


















Equilibrium conditions

An equilibrium price p∗is such that for each good i = 1, ...,n:Ei (p∗)≤ 0

p∗i ≥ 0

p∗i Ei (p∗) = 0

The first condition requires that there cannot be excess demand.

The second condition requires prices to be non-negative.

The third condition tells that if there is excess supply, then the pricemust be 0 (free good).




p∗i ≥ 0

p∗i Ei (p∗) = 0







p∗i ≥ 0

p∗i Ei (p∗) = 0







p∗i ≥ 0

p∗i Ei (p∗) = 0





Properties of the excess demand (1)

Property 1. The excess demand is homogeneous of degree 0.

As a consequence, for each prices p̃, we can define normalized prices pas follows:

p1 = p̃1p̃1+...+p̃n...

pi = p̃ip̃1+...+p̃n...

pn = p̃np̃1+...+p̃n

Define the set of all possible normalized prices as

J :=

{p ≥ 0

∣∣∣∣∣ n

∑i=1

pi = 1

}





p1 = p̃1p̃1+...+p̃n...

pi = p̃ip̃1+...+p̃n...

pn = p̃np̃1+...+p̃n


J :=

{p ≥ 0

∣∣∣∣∣ n

∑i=1

pi = 1

}





p1 = p̃1p̃1+...+p̃n...

pi = p̃ip̃1+...+p̃n...

pn = p̃np̃1+...+p̃n


J :=

{p ≥ 0

∣∣∣∣∣ n

∑i=1

pi = 1

}



Property 2. Walras’ Law. Assume household’s preferences arerational and satisfy greed and firms are profit maximizer and privatelyowned by households. Then for any price p the excess demandfunctions are such that

n

∑i=1

piEi (p) = 0

Rearranging, this gives pnEn (p) + ∑n−1i=1 piEi (p) = 0, or:

En (p) =− 1pn

n−1

∑i=1

piEi (p)



Property 2. Walras’ Law. Assume household’s preferences arerational and satisfy greed and firms are profit maximizer and privatelyowned by households. Then for any price p the excess demandfunctions are such that

n

∑i=1

piEi (p) = 0

Rearranging, this gives pnEn (p) + ∑n−1i=1 piEi (p) = 0, or:

En (p) =− 1pn

n−1

∑i=1

piEi (p)


Existence theorem

ExistenceIf each excess demand function is continuous function from J (the set ofnormalized prices) to the real line and is bounded below, then there existsp∗ ∈ J that is an equilibrium price vector.

Sufficient conditions for the continuity of the excess demand functionsare the strict concavity of the production functions Φf and the strictquasi-concavity of the utility functions Uh. [Concavity andquasi-concavity are sufficient for existence, but the excess demandscould be multivalued, i.e. correspondences.]

Existence does not mean uniqueness; moreover, even if an equilibriumexists, it might be that the economy will not converge there.


Existence theorem





Existence theorem





Outline



From equilibrium prices to a competitive equilibrium

The existence of equilibrium prices means that for each good i :Ei (p∗)≤ 0

p∗i ≥ 0

p∗i Ei (p∗) = 0

Given prices p∗, the problem of each household and each firm iswell-defined: let each choose freely to maximize utility and profitsrespectively.Let a∗ = ([x∗] , [q∗]) be the corresponding allocation. We can writethat:

p∗→ a∗.



The existence of equilibrium prices means that for each good i :Ei (p∗)≤ 0

p∗i ≥ 0

p∗i Ei (p∗) = 0

Given prices p∗, the problem of each household and each firm iswell-defined: let each choose freely to maximize utility and profitsrespectively.Let a∗ = ([x∗] , [q∗]) be the corresponding allocation. We can writethat:

p∗→ a∗.



Then ([x∗] , [q∗] ,p∗) is a competitive market allocation.

Moreover, the material balance condition is satisfied, thus it is also acompetitive equilibrium.



Then ([x∗] , [q∗] ,p∗) is a competitive market allocation.

Moreover, the material balance condition is satisfied, thus it is also acompetitive equilibrium.


The role of prices (1)

Define the attainable set as the set of all aggregate quantities ofgoods that can be made available for consumption:

A := {x |x ≤ q +R and Φ(q)≤0}

or all [x ] such that

nh

∑h=1

xhi ≤

nf

∑f =1

qfi +

nh

∑h=1

Rhi for each i

andΦf(qf)≤ 0 for each f .


The role of prices (2)Define the better-than-x* set as the set of all aggregate quantities ofgoods that would give the household at least the same utility as in x∗:

B :={

x∣∣∣Uh

(xh)≥ Uh

(x∗h)

for each h}.

Let the aggregate expenditure/income be:

y :=n

∑i=1

p∗i x∗i

and define the hyperplane

X y :=

{x

∣∣∣∣∣ n

∑i=1

p∗i xi = y

}.

The two sets A and B are tangent in x∗ and separated by thehyperplane X y .



B :={

x∣∣∣Uh

(xh)≥ Uh

(x∗h)

for each h}.


y :=n

∑i=1

p∗i x∗i


X y :=

{x

∣∣∣∣∣ n

∑i=1

p∗i xi = y

}.




B :={

x∣∣∣Uh

(xh)≥ Uh

(x∗h)

for each h}.


y :=n

∑i=1

p∗i x∗i


X y :=

{x

∣∣∣∣∣ n

∑i=1

p∗i xi = y

}.




B :={

x∣∣∣Uh

(xh)≥ Uh

(x∗h)

for each h}.


y :=n

∑i=1

p∗i x∗i


X y :=

{x

∣∣∣∣∣ n

∑i=1

p∗i xi = y

}.



The value of consumption and production

Separating hyperplaneIf A and B are convex sets, then there are prices p∗ and a consumptionvector x∗ such that:

n

∑i=1

p∗i xi ≤ y for each x ∈ A

andn

∑i=1

p∗i xi ≥ y for each x ∈ B

where y := ∑ni=1 p∗i x

∗i .

Interpretation: x∗ maximizes the value of aggregate income over A(attainable alternatives) and minimizes the aggregate cost over B (thebetter-than-x* set).


The value of consumption and production

Separating hyperplaneIf A and B are convex sets, then there are prices p∗ and a consumptionvector x∗ such that:

n

∑i=1

p∗i xi ≤ y for each x ∈ A

andn

∑i=1

p∗i xi ≥ y for each x ∈ B

where y := ∑ni=1 p∗i x

∗i .

Interpretation: x∗ maximizes the value of aggregate income over A(attainable alternatives) and minimizes the aggregate cost over B (thebetter-than-x* set).


Documents

Part1 P.Piacquadio€¦ · [email protected] September24,2014 P. Piacquadio ([email protected]) Micro 3200/4200 September 24, 2014 1 / 26. Outline 1 GeneralEquilibrium