General competitive equilibrium. General equilibrium: How does an idealized private ownership competitive economy works ? l goods (indexed by j )

General competitive equilibrium

General equilibrium: How does an idealized private ownership

competitive economy works ? l goods (indexed by j) n individuals (households) (indexed by i) k firms (or technologies) (indexed by f) firm f’s technology: a production set Yf l that

is closed, irreversible, convex and satisfies the possibility of inaction, impossibility of free production, and free disposal.

All goods are private (rival and excludable). They are privately owned. i

j 0: quantity of good j initially owned by household i


competitive economy works ? (2)

1 if 0 : share of firm f owned by household i

Each firm is entirely owned (i if = 1 for all f)

Xi l+: Consumption set of household i

(convex and closed) i: preferences of household i (reflexive,

complete, transitive, continous, locally non-satiable and convex binary relation on Xi).



An economy = (Yf,Xi,i,if,i

j), i =1,…,n, j =1,…,l and f = 1,…k.

Economic problem: finding an allocation of the l goods accross the n individuals.

Some allocations are feasible, some are not.

A(): the set of all allocations of goods that are feasible for the economy .



A() is defined as follows:

k

f

fj

n

i

ij

n

i

ij

ffii jyxtskfforYyniforXx111

..,...,1:,...,1

In words, A() is the set of bundles of goods that could be consumed in the economy given its technological possibilities, and the initial available resources (under the assumption that these resources are publicly owned)

A nice geometrical depiction of the set of feasible allocations: the Edgeworth box Suppose Yf = {0l} for all f (nothing is

produced) Then is an exchange economy. A() in this case can be defined by:

jxniforXxn

i

ij

n

i

ij

ii

11

:,...,1

If l = n =2, we can represent the bundles thatsatisfy this weak inequality at equality on the

following diagram

The Edgeworth Box

Individual 1

individual 2

x22

x11

x12

x21

x

2 =1

2 + 22

1= 11 + 2

1

The Edgeworth Box

Individual 1

individual 2

2

1

x

Pareto efficiency

Some feasible allocations of goods involve waste.

Some feasible allocations of goods do not exhaust the existing possibilities of mutual gains (called « win-win » situations in ordinary language)

Some feasible allocations of goods are not Pareto-efficient!

Pareto efficiency

Definition: an allocation xij A() (for i = 1,

…,n and j = 1,…,l) is Pareto-efficient in A() if, for any other allocation zi

j A(), having zh xh for some individual h must imply that xg zg i = 1,…,n. some individual g.

In words, an allocation xij A() (for i = 1,…,n

and j = 1,…,l) is Pareto-efficient in A() if it is impossible to find an allocation in A() that everybody weakly prefers to xi

j and that at least one person strictly prefers to xi

j

Pareto-efficiency in an Edgeworth Box

1

2

x22

x11

x12

x21

x

y

2

1

z


1

2

x22

x11

x12

x21

x

y

z

2

1

z is not Pareto-efficient


1

2

x22

x11

x12

x21

x

y

2

1

Allocations in this zone are unanimouslypreferred to z

z


1

2

x22

x11

x12

x21

x

y

2

1

Allocation y (among other) is unanimouslypreferred to z

z


1

2

x22

x11

x12

x21

x

y

2

1

Allocation y is Pareto-efficientz


1

2

x22

x11

x12

x21

x

y

2

1

So is x !

z


1

2

x22

x11

x12

x21

x

y

2

1

So are all theallocations onthe blue locusz

Pareto efficiency

A minimal normative requirement. An inefficient allocation is unstatisfactory. Yet Pareto-efficiency is hardly a sufficient

requirement. There are many Pareto-efficient allocations,

and some of them may involve significant inequality

As Amartya Sen put it « a society may be Pareto-efficient and perfectly disgusting!

General Competitive equilibrium

What happens when all households and all firms take their decisions individually, taking as given a prevailing set of prices ?

Given prices, each firm chooses a production activity that maximizes its profits.

GIven prices, each household chooses a bundle of l goods that it most prefer.

Prices are such that these choices are mutually consistent (supply equal demand on all markets).


Here is a formal definition. A General Competitive Equilibrium (GCE) for

the economy = (Yf,Xi,i,ik,i

j), i =1,…,n, j =1,…,l and f = 1,…k is a list (p*,xi*,yf*) with p* l

+ , xi* Xi for i =1,…,n, yf* Yf

for f =1,…k such that:

)1(),...,,,...,,,...,(

),...,,,...,,,...,(

11**

1*

11**

1*

ik

iil

il

iii

ik

iil

il

ii

ppBzzx

ppBx



the economy = (Yk,Xi,i,ik,i




)2(1

*

1

** fl

jjj

l

j

fjj Yyypyp



the economy = (Yk,Xi,i,ik,i




)2(1

*

1

** fl

jjj

l

j

fjj Yyypyp

and



the economy = (Yf,Xi,i,ik,i




)3(1 1

*

1

* jyxn

i

k

f

fj

ij

n

i

ij


Condition 1) says that given prices, and the budget constraint that these prices define (given initial endowments and ownerships of firms), household i chooses a bundle of goods that it most prefer in its budget set.

Condition 2 says that given prices firm k chooses a production activity in its production set that maximizes its profits.

Condition 3 says that choices made by firms and consumers are all mutually consistent (on every market, the demand for the good is never superior to the amount of good availabe (both as the result of production and initial endowments).

General-equilibrium in an Edgeworth Box (no production)

1

2

12

11

21

22


1

2

12

11

21

22


1

2

x22

x11

12

11

21

22

-p*1/p*

2

(p*11

1+ p*

212)/p*

2


1

2

x22

x11

12

11

21

22

-p*1/p*

2

(p*11

1+ p*

212)/p*

2


1

2

x22

x11

12

11

21

22

-p*1/p*

2

(p*11

1+ p*

212)/p*

2

x2*1

x1*1


1

2

x22

x11

12

11

21

22

-p*1/p*

2

(p*11

1+ p*

212)/p*

2

x2*1

x1*1

x1*2 x2*

2

Excess demand correspondance

Condition 3 defines what is called the « excess demand correspondance » Z: l

+ l as follows:

n

i

k

f

fin

i

Mi pypXpZ1 1

*

1

)()()(

Where XMi(p) is the Marshallian demand correspondance of household i at prices p and yf*(p) is the supply/demand correspondanceof firm f at prices p

Walras Law Theorem: if consumers preferences satisfy

local non-satiation, then, for all elements Z in the excess demand correspondance

l

jljj ppZp

11 0),...,(

Proof: Under local non-satiation, one has for any household i, and any element xMi of its Marshallian demand correspondance:

l

j

k

f

l

j

fjj

ik

ijj

l

j

Mijj pypppxp

1 1 1

*

1

)()(

Walras Law Theorem: if consumers preferences satisfy

local non-satiation, then, for all elements Z in the excess demand correspondance

l

jljj ppZp

11 0),...,(

Proof: summing this equality over all individuals yields (exploiting the fact that firms’ shares sum to 1)

l

j

kjj

n

i

n

i

l

j

ijj

l

j

Mijj pypppxp

1

*

1 1 11

)()(

A corrolary of Walras Law

Theorem: if an element Z of the excess demand correspondance satisfies Walras law and if Zj(p) 0 for all goods j, then pg = 0 for any good g for which Zg(p) < 0.

Proof: obvious

Interpretation: if the market for a good is in strict excess supply at a CGE, then the price of this good must be zero (example: sand, stones, etc.)

Consequence of this corrolary

If (p*,xi*,yk*) is CGE for an economy, then for every good g for which p*g > 0 one must have Zg(p*) = 0.

We are going to use this to prove the existence of CGE for any economy satisfying our assumptions.

Establishing the existence of CGE has been one of the major achievement of the 20th century mathematical economics (finalized in fifeties through the work of Arrow and Debreu.

Argument is based on the fact that certain functions admit Fixed Points.

Fixed points Many existence theorems in mathematical economics

and game theory (Nash equilibrium, GCE, etc.) are consequences of mathematical theorems known as « fixed point » theorems.

Two such theorems are particularly useful: Brouwer’s fixed point theorem (that deals with functions) and Kakutani’s fixed point theorem that deals with correspondances.

In order to understand these theorems, we must first understand what is a fixed point.

Definition: Given a function f: A A , we say of an element a A that it is fixed point of f if f(a) = a

Some functions do not admit fixed points. For example, the function f that assigns to every alive

individual whose father is alive this father does not have fixed point (because nobody is his or her own father)

Brouwer’s fixed point theorem The mathematician Brouwer established

(in 1912) a theorem guaranteeing that a (real-valued) function admits a fixed point.

Brouwer’s fixed point Theorem: Let A be a convex and compact subset of k and let f: A A be a continuous function. Then, there exists an element a A that is a fixed point of f

Let us illustrate the theorem when A is a convex and compact subset (and therefore an interval) of .

Brouwer fixed point theorem

A

A

x

y

y = x

f


A

A

x

y

y = x

Fixed points

f

Brouwer fixed point Theorem

A

A

x

y

f

The assumptions of Brouwer’s theorem

are all important


A

A

x

y

f

For example thefunction f

does not have any fixed point


A

A

x

y

f

but f is not a function

from A to A


A

A

x

y

f

This function f (that is not continuous)

does not have fixedpoint either!


A

A

x

y

f

Similarly, if A isopen above a

continuous function like f does not have

fixed points

Fixed points Brouwer’s fixed point theorem applies to functions. Yet

the CGE can be viewed as the fixed point of a correspondance of price adjustment (based on excess demand).

Kakutani (1941) has generalized Brouwer’s fixed point theorem to correspondances.

Kakutani fixed point theorem: Let A be a convex and compact subset of l and let C: A A be correspondance upper-semi continuous. Then, there exists an element a* A that is a fixed point of C (and that is therefore such that a* C(a*)).

John Nash in 1950 has used this theorem to show the existence of a Nash equilibrium in non-cooperative games.

Debreu (1959) has used it as well in his proof of the existence of CGE.

Upper Semi-continuity ?

A

A

x

y

C

The correspondence C is upper semi-continuous

Upper semi-continuity ?

A

A

x

y

C

The correspondence C is upper semi-continuous

Upper semi-continuity ?

A

A

x

y

C

The correspondence C is not

Upper semi-continuous

Berge (1959) Maximum Theorem Theorem: Let A and B be two subsets of

k with A compact and let : AB be a continuous function. Then, the correspondence C: B A defined by:

),(maxarg)( babCa

is upper semi continuous

Applies Berge Maximum Theorem (1) To the profit maximization program.

l

jjj

Yyyl ypppy

fl 1),...,(

1*

1

maxarg),...,(

Here, set A of the theorem is the production set Yk

Problem: Yf is not compact (it is closed but possibly unbounded).

Solution: Maximize profits on the set Yf {y l: y -} \ l

- - (exclude plans that use more inputs than what is initially available and/or that uses input without producing any output).

Applies Berge Maximum Theorem (2) To the consumer’s utility maximization

program:

),...,(maxarg),...,( 1),...,,.,...,,,...,(),...,(

11111

li

ppBxxl

M xxUppXik

iil

ill

Here, set A of the theorem is the budget set (that is compact)

Existence of a GCE (Arrow-Debreu) (1) An economy = (Yf,Xi,i,i

f,ij), i=,…,n,

j =1,…,l and f = 1,…k admits a GCE if:

Yf satisfies closedness, irreversibility, impossibility of free production, possibility of inaction, free disposal and convexity for f = 1,…k

Xi is a closed and convex subset of l+

containing a bundle xi distinct from (i1, …,i

l ) such that (i

1, …,il ) > xi for i =,…,n

i is reflexive, transitive, complete, continuous, locally non-satiable and convex for i =,…,n.

Existence of a GCE (Arrow-Debreu) (2) Proof: We find convenient to limit our quest

for equilibrium prices to the set: Sl-1 = {(p1,…,pl) l

+ : p1+…+pl = 1} This does not involve any loss of generality

since agents’ behavior at given prices is unaffected by proportional changes in all prices

Hence, facing prices (p1,…,pl) is equivalent to facing prices (p1/(p1+…+pl),…,pl/(p1+…+pl))

Sl-1 is clearly compact (it is called a « simplex »)

Define the correspondance A: Sl-1 Sl-1 as follows:

Existence of a GCE (Arrow-Debreu) (3) A(p1,…,pl)= for some element Z of the Excess

demand correspondance This correpondance may be viewed as a

« price adjustment » rule. For any initial configuration of prices, it finds a new set of prices that maximizes the value (at those prices) of any particular element of the Excess demand correspondance)

The set Sl-1 is compact The correspondance A is upper semi-

continuous (as is the excess demand correspondance by Berge Maximum theorem)

Hence the correspondance A admits a fixed point (p*1,…,p*l)

This fixed point of A is a GCE of the underlying economy.

Details: see Debreu (1959).

Existence of a GCE (Arrow-Debreu) (3)

1st Welfare theorem If (p*,xi*,yf*) is a GCE for the economy =

(Yf,Xi,i,ik,i

j), i =1,…,n, j =1,…,l and f = 1,…k, and if i is reflexive, complete, transitive, and locally non-satiable, then the allocation xi*, (for i=1,…,n) is Pareto-efficient in A().

Proof: by contradiction, suppose that (p*,xi*,yf*) is a

GCE for the economy = (Yf,Xi,i,ik,i

j), i =1,…,n, j =1,…,l and f = 1,…k, but assume that xi*, is not Pareto-efficient in A().

1st Welfare theorem If (p*,xi*,yf*) is a GCE for the economy =

(Yf,Xi,i,ik,i

j), i =1,…,n, j =1,…,l and f = 1,…k, and if i is reflexive, complete, transitive, and locally non-satiable, then the allocation xi*, (for i=1,…,n) is Pareto-efficient in A().

Proof: by contradiction, suppose that (p*,xi*,yf*) is

a GCE for the economy = (Yf,Xi,i,ik,i

j), i =1,…,n, j =1,…,l and f = 1,…k, but assume that xi*, is not Pareto-efficient in A(). Hence, there exists an allocation xi (for i=1,…,n) in A() such that:

1st Welfare theorem

*

*

hxhhx

andiixiix

For at least one

household h (1)

Since xi A(), there exists productie activities yf Yf (for f=1,…k) such that:

(2)

Condition (1) implies that, for all household i, one has:

k

f

fj

n

i

ij

n

i

ij jyx

111

1st welfare theorem

And, for at least one household h, one has:

k

ff

if

l

j

ijj

l

j

ijj ppxp

1

*

1

*

1

* )(

Summing inequalities (3) over all households (with (4) holding for at least one of them) yields:

k

ff

hf

l

j

hjj

l

j

hjj ppxp

1

*

1

*

1

* )(

(3)

(4)

1st welfare theorem

since, for all f

k

fk

n

i

n

i

l

j

ijj

l

j

ijj ppxp

1

*

1 1 1

*

1

* )(

Moreover, since yf* maximises firm f profits

at prices p*, one has, for any such firm:

n

i

if

1

1

(5)

l

j

l

j

fjj

fjjf ypypp

1 1

****)( (6)

1st welfare theorem

k

f

l

j

fjj

n

i

n

i

l

j

ijj

l

j

ijj yppxp

1 1

*

1 1 1

*

1

*

Substituting (6) into (5), one gets:

Which contradicts the requirement that Inequality (2) holds for every good j .

QED.

Meaning of the theorem

If all goods that matter for humans and all technologies that enable the production of certain goods out of others are privately owned, then the free working of the market – provided that it operates under perfect competition – leads to a Pareto efficient resources allocation.

This theorems points naturally to the limitation of the competitive market

mechanism Some important goods should not or can not

be privately appropriate (non-rivalry or impossibility of exclusion).

Some markets can ot be perfectly competitive because the efficient scale of production is large relative to the market)

Some markets (insurance among others) can not appear because of severe problems of information asymetries (moral hazard and/or adverse selection)

Limits of this theorem Efficiency is not everything! There are many efficient allocations. One can be efficient while being « unjust » (this

requires of course a definition of « justice ») The 2nd welfare theorem answers in part to these

limitations. 2nd welfare theorem: any Pareto-efficient

allocation can be obtained as the result of the integrated funtionning of competitive markets provided that lump sum redistribution of the purchasing power among households be initially performed.

2nd wefare theorem If = (Yf,Xi,i,i

k,ij), pour i =1,…,n, j =1,…,l and f = 1,…k is

an economy satisying all assumptions that guarantee the existence of a CGE and if the allocation (xi*) i=1,…,n is Pareto efficient in A(), then there are (possibly negative) lump sum taxes Ti for i =1,…,n, satisfying T1 +…+ Tn = 0, prices p1*,…, pl* +

l and productive activities yf* Yf (for f = 1,…k) such that:

)1(),...,(

..

),...,(

1

***1

1

**

1

*

*

1

***1

1

**

1

**

K

fil

fif

l

j

ijj

l

jjj

iii

k

fil

fif

l

j

ijj

l

j

ijj

Tpppzp

tsXzzx

Tpppxp





)2(

1

*

1

** fl

jjj

l

j

fjj Yyypyp





)3(

1 1

*

1

* jyxn

i

k

f

fj

ij

n

i

ij

1

2

x22

x11

12

11

21

22

-pe1/pe

2

(pe11

1+pe

212)/pe

2

2nd welfare theorem in an Edgeworth Box

(pe12

1+pe22

2)/pe2

x1*1

x2*1

-p*1/p*

2(p*

121+

p*22

2 - T2)/p*

2

(p*11

1+ p*

212 -

T1)/p*2

The 2nd welfare theorem rides on much more assumptions than the first

For example, it may not hold if preferences are not convex.

Let us illustrate this graphically.

1

2

x22

x11

12

11

21

22

The 2nd welfare theorem requires convexity of preferences

(p*11

1+ p*

212 -

T1)/p*2

-pe1/pe

2

(p*12

1+ p*

222 -

T2)/p*2

Documents

General competitive equilibrium. General equilibrium: How does an idealized private ownership competitive economy works ? l goods (indexed by j )