03 Micro Walras

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Part III:

Walrasian Theory


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Literature

Geoffrey A. Jehle, Philip J. Reny: Advanced

Microeconomic Theory, 2nd Ed., PearsonInternational, 2001.

2

Hall R. Varian: Microeconomic Analysis, 3rdEd., W. W. Norton & Company, 1992.

Andreu Mas-Colell, Micheal D. Whinston undJerry R. Green: Microeconomic Theory,Oxford University Press, 1995.


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Overview

1. Introduction

2. Model

3. Normative Analysis

3

Second Welfare Theorem

4. Positive Analysis

Existence Structural characteristics


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Introduction

In Part II of this lecture series we looked at

partial analysis, where makets wereconsidered in isolation.

Where the income effect and repercussions

4

in other markets are important, partialanalysis can yield false results.

The following example from Mas-Colell et al.

(1995, pp. 538540) makes this evident:


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IntroductionWho bears the tax burden?

1 country, Nidentical towns, Nidentical firms

5

,to 1

One firm per town

Production function, with z= labor input:

( ) ( ) ( )mit ' 0 und '' 0 f z f z f z>


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Introduction

Total supply of labor is M; and supply is

inelastic.

6

wn is the wage in town n

Complete factor mobility implies that

1 ..... nw w w= = =


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Introduction Each firm employs M/Nunits of labor.

The following wage is the product of the optimization

condition for firms:

7

Town 1 decides to tax its firm.

( )Marginal Cost

Marginal Revenue

' MNw f=


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Introduction

Owing to tax t, the optimization condition for

firm 1 in town 1 is new.

( )

' MN

w t f+ =

8

Who will actually pay the tax? The worker or

the firm?

Marginal Revenue


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Introduction

Partial analysis

Partial analysis investigates this questionunder the assumption that wages remain

9

. .,

Because of complete factor mobility, thewage in town 1 cannot fall, thus

2 ..... nw w w= = =

1w w=


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Introduction

Therefore, the whole tax must be carried by

firm 1.

10


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IntroductionGeneral analysis

We now look at the general equilibrium across the

labor markets of all the towns. As a result ofcompetition, the equilibrium wage rate must be such

11

as

where the wage w(t) depends on taxation in town 1.

The firms in towns 2, ... ,Nrespectively employ z(t)and firm 1, z1(t) units of labor.

( )1 2 ..... nw w w w t = = = =


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Introduction

The following equilibrium conditions hold:

( ) ( ) ( ) 11N z t z t M + =

12

Labor supplyLabor demand

( )( ) ( )

( )( ) ( )1

'

'

f z t w t

f z t w t t

=

= +


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Introduction

How does the wage w(t) react when the tax t

changes?

13

respect to t, we obtain:

( ) ( ) ( )

( )( ) ( ) ( )

( )( ) ( ) ( )

1

1 1

1 ' ' 0 (1)

'' ' ' 0 (2)

'' ' ' 1 0 (3)

N z t z t

f z t z t w t

f z t z t w t

+ =

=

=


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Introduction

We can now substitute from(2) for z'(t) and

from (3) for z1'(t) in (1).

( )( ) ( )' ' 1

1 0 (4)'' ''

w t w t N

+ + =

14

If we now evaluate (4) at t= 0, we obtain

since z(0) = z1(0) = M/N.

1

( ) ( ) ( )1 ' 0 ' 0 1 0 (5) N w w + + =


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Introduction

Solving equation (5) for w'(0) gives

( ) 1' 0w =

15

The equilibrium wage thus falls in all Ntowns.

The reduction is smaller, the more townsthere are.


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Introduction

Now we still do not know who will pay the tax.

We only know that the workers carry a portionof the tax burden.

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Let a firms profit function be .

Then aggregate profit is:

( )w

( ) ( )( ) ( )( )1N w t w t t + +


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Introduction The change in the aggregate profit is

If we evaluate this change when t= 0, we

( ) ( )( ) ( ) ( )( ) ( )( )1 ' ' ' ' 1N w t w t w t t w t + + +

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Aggregate profits do not change!

Only the workers pay!

( )( )1 1

' 0 1 0N

wN N

+ + =


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Introduction Although the partial equilibrium approximation

is correct, as far as getting prices and wagesabout right, it errs by just enough and in justsuch a direction that the conclusion of the tax

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incidence analysis based on it is completelyreversed.


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Introduction This example has shown that it is often

important to undertake a total analysis of themarkets.

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That is, we have to investigate all markets andtheir mutual dependencies simultaneously.

This is the aim and purpose of generalequilibrium theory (Walrasian Theory).


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IntroductionUse of the theory

General equilibrium analysis offers a framework for

investigating the effects of exogenous changes such

20

, , ,

or politics on quantities and prices in all markets.

It is particularly important that general equilibrium

effects are also taken into consideration whenevaluating fiscal or structural policy measures.


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IntroductionExamples

Effects of an eco-tax

21

ec s o a or mar e po c es

Effects of a oil price shock


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IntroductionAreas of application

Public Finance

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Macroeconomics

Finance

Trade theory


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Introduction Historical description of general equilibrium

theory in a nutshell

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that the wealth of the nation was based onthe division of labor and the achievement of

economic subjects individual interests .

The price mechanism directed by an invisible

hand brought about intelligent coordination.


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Introduction Adam Smiths book, An Inquiry into the

Nature and Causes of the Wealth of Nations(1776), founded economic liberalism.

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s centra t es s was t at se - nterestcreated (state) welfare: It is not from thebenevolence of the butcher, the brewer or thebaker that we expect our dinner, ... We

address ourselves not to their humanity but totheir self-love, and never talk to them of ourown necessities but of their advantages.


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Introduction L. Walras (1886) was the first to formulate a

mathematic model taking up Adam Smithsideas of the invisible hand.

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According to Walras an equilibrium is a pricevector pthat brings supply and demand in allmarkets into balance.


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Introduction Kenneth Arrow and Gerard Debreu applied

Kakutanis fixed-point theorem in the 1950sin order to prove the existence of a Walrasianequilibrium subject to certain assumptions.

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Already before they were able to substantiate

the existence of Walrasian equilibria, they

proofed the first and second fundamentaltheorems of welfare theory.


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The Model

These two fundamental theorems of welfaretheory constitute the theoretical foundation of

the social market economy.

27

We shall give a short introduction to the

general equilibrium model according to

Debreu (Theory of Value, 1959).


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The ModelImportant assumptions of the model as presented here

The number of goods is given.

-

28

.

Household preferences are given.

The allocation of property rights is given (i.e., they areuniquely defined, implicitly assuming there is an efficientlegal system).


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The Model

The agents exchange goods at prices thatthey regard as given.

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Exchange is central and there are notransaction or information costs (frictionless

economy).

Prices are constituted such that all markets

are cleared.


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The ModelGoods

There arej= 1,, Ngoods.

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Goods are perfectly divisible.

There is a market for every good (completemarkets).


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The ModelPrices

Let pjbe the price of goodj.

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We then call a pricevector.

( )1,....,N

N p p p

+=


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The ModelProducers (firms)

There are f= 1,, F producers.

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determined by the technology .

The following sign convention applies for theproduction plans :

Inputs are negative, outputs are positive.

NfY

( )1,...,

f f fN f

y y y Y =


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The Model

We assume that producer fmaximizes his

profit , given his technologicalpossibilities.

( )f p

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That is, he solves the following problem.

( ) fiN

i

i

Yy

f

Yy

f

Yy

ypyppffffff

=

==

1

maxmaxmax


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The ModelConsumers (Households)

There are h= 1,,H households.

34

Household hstarts with the initial endowment:

The sum of initial allocations

is the total allocation for the economy.

( )1,...,N

h h hN e e e

+=

1

H

h

h

e e=

=


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The Model The set is the set of all feasible

consumption bundles of consumer h.

N

hX

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Each household possesses a utility function.

( )1,...,h h hN h x x x X =

:h hu X


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The Model Firms belong to the consumers and the profits from

production are split among the consumers.

Let hf be the portion of the h-th consumers share ofprofit from the f-th firm.

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It holds that:

Let the vector of the profit shares of the h-thconsumer be

[ ]1

0,1 und 1H

hf hf

h

=

=

( )1,...,h h hF =


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The Model

Each individual households budget is

composed of the value of their initialendowment as well as their shares from

com anies rofits:

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( )1

F

h h hf f

f

p x p e p=

= +


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The Model

Households maximize their utility subject to their

budget constraint.

The decision roblem is thus:

38

( ) ( )1

max s.t.h h

F

h h h h hf f x X

f

U x p x p e p

=

= +


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The ModelDefinition (Allocation): A list

of consumption plans and production plans is

( ) ( )1 1, ,.., ; ,..,H Fx y x x y y=

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called an allocation if

( )

( )

1

1

,.., for all 1,..,

,.., for all 1,..,

h h hN h

f f fN f

x x x X h H

y y y Y f F

= =

= =


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The Model

Definition: An allocation is feasible, if thefollowing holds:

40

===

+F

f

f

H

h

h

H

h

h yex111


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The Model

Feasibility simply means that for eachgood the quantity consumed cannot be

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.


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The ModelDefinition (Walrasian equilibrium). An

allocation (x*; y*) with price vector p* is aWalrasian equilibrium if the following holds:* *1. arg max 1,..., y p y f F =

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( )

( )

*

* * * *

1

* * *

1 1 1

2. arg max s.t.

1,...,

3.

f f

h h

y Y

h h hx X

F

h h hf f f

H H F

h h f

h h f

x u x

p x p e p y h H

x e y

=

= = =

+ =

= +


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The ModelIn words

In a Walrasian equilibrium all decision makerstake the price vector as given.

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Firms maximize their profits and householdstheir utility.

The price vector is such that for each goodsupply = demand.


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The ModelQuestions

Normative: What welfare characteristics does aWalrasian e uilibrium have was Adam Smith

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right?)

Positive: Existence and characteristics

Empirical: Which real markets fit this model?


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The ModelQuestions

Normative: Welfare properties

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Positive: Existence and comparative statics

Empirical: does the model fit the data


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Overview

Behavioralhyothesis

Exogenousdata

Endogenousdata

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Households Utilitymaximization

EndowmentPreferences

Prices

FirmsProfitmaximization

Technology Prices


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The ModelExample: Barter economy

We will now calculate the Walrasianequilibrium for a barter economy in which

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1 2

households.


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The ModelExample: Barter economy

Initial endowment

1 0 and 0 1e e e e e e= = = =

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Utility functions

( ) ( )( )

1 1

1 11 12 11 12 2 21 22 21 22, and , ,, 0,1

u x x x x u x x x x

= =


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The ModelExample: Edgeworth-Box with endowment e

x12x21

2ee21 = e1- ee11

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x111 x22ee11

ee22 = e2- ee12ee12

~2

~1


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The ModelCalculation of individual demand functions

Household 1 solves the following problem:

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11 12 1 11 12 11 12,

1 11 2 12 1

1 1 11 2 12

,

s.t.

where

x x

ax u x x x x

p x p x b

b p e p e

=

+ =

= +


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The Model Lagrange:

( )111 12 1 1 11 2 12L x x b p x p x = +

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( )

1 1

11 12 1

11

11 12 2

12

1 1 11 2 12

0

1 0

0

L x x p

x

L x x px

Lb p x p x

= =

= =

= =


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The Model Marginal rate of substitution is equal to the

relative price12 1

11

x pMRS

x

= =

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The same optimization principle applies to the

other household.

22 12

21 21

x pMRS

x p= =


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The Model We obtain individual demands via the budget

constraint:

( )[ ]1 11 2 12

11 1 2

1

,p e p e

x p pp

+=

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For the other household we obtain:

( )

1 11 2 12

12 1 22

,p e p e

x p pp

+=

( )[ ]

( )( )[ ]

1 21 2 22

21 1 2

1

1 21 2 22

22 1 2

2

,

1,

p e p e x p p p

p e p e x p p

p

+

=

+=


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The ModelCalculation of equilibrium prices

Aggregate demand = aggregate supply

( ) ( )11 1 2 21 1 2 11 21, ,x p p x p p e e+ = +

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Aggregate demand is homogeneous of degree zero. We cantherefore normalize a price.I choose p1 = 1:

1 11 2 12 1 21 2 22

11 21

1

2

1

1

e ep

p

p

= +

+ =

2 1p + =


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The Model Equilibrium prices are thus:

1 211,p p = =

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Equilibrium consumption is thus:

( ) ( )

( ) ( )

11 1 2 12 1 2

21 1 2 22 1 2

, ; ,

, 1 ; , 1

x p p x p p

x p p x p p

= =

= =


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NORMATIVE ANALYSIS

N i A l i


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Normative Analysis

Normative analysis is concerned with the

question of how allocations should beevaluated.

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Here, we investigate the welfare properties ofWalrasian equilibria.

Welfare evaluations are always based on

subjective judgments.

N i A l i


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Normative Analysis

A central question of economics is: How

should goods be allocated amonghouseholds?

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We now consider 3 alternatives Welfare functions

Voting

Pareto Criterion

N ti A l i


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Normative Analysis

Alternative 1: Welfare functions We define a welfare function:

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= u1,,uH

and look for the allocation that maximizes

welfare.

N ti A l i


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Normative Analysis

Examples of welfare functions W(u1, u2)

1. The utilitarian welfare function:

W(u1, u2) = u1 + u2

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Characteristics:

(a) Symmetry: W(u1, u2) = W(u2, u1)

Symmetry has the advantage that welfare does notdepend on the name of the consumer, but rather oneveryone being considered in the same way.

N ti A l i


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Normative Analysis

(b) Always give the most to the individual with the greatestmarginal utility. This function awards something to the

individual who can produce greater utility with thegoods.

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.

W(u1, u2) = min{u1, u2}

Characteristics:

(a) Symmetry: W(u1, u2) = W(u2, u1)

(b) Give to the one with the least utility.

Normati e Anal sis


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Normative Analysis

Criticism of welfare functions

Who chooses W(u1,, uH)?

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An interpersonal comparison of utility is required.

Individuals have an incentive to hide their truepreferences if they see an advantage in doing so.

Normative Analysis


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Normative Analysis

Interpersonal comparison of utility requires a

cardinal utility concept. The possibility of this,however, is questionable and is the reason why anordinal utility concept has become accepted.

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Normative Analysis


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Normative Analysis

Alternative 2: Voting

Derivation of a welfare function based on individualhousehold preferences decided by means of majorityvoting choices.

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Condorcet Paradox

Arrows Impossibility Theorem

Normative Analysis


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Normative Analysis

Condorcet Paradox

Condorcets paradox illustrates that the familiarmethod of majority voting can fail to satisfy thetransitivit re uirement.

65

Let there be 3 social states a, b, c:

Player 1 a b c

Player 2 b c a

Player 3 c a b

( ) ( ) ( )cubuau 111 >>

( ) ( ) ( )aucubu 222 >>

( ) ( ) ( )buaucu 333 >>

Normative Analysis


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Normative Analysis

Voting:

a vs. b: Players 1 and 3 find a better.Thus W(a) > W(b)

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Thus W(b) > W(c)a vs. c: Players 2 and 3 find c better.

Thus W(c) > W(a)

Thus W(a) > W(b) > W(c) > W(a) which is acontradiction.

Normative Analysis


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Normative Analysis

Arrows Impossibility Theorem

Arrows impossibility theorem shows the impossibilityof aggregating individual preferences (finding a votingrule) so as to establish a social preference order that

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is free of contradictions.

It states that:If there are at least three social states, then there is

no social welfare function that satisfies the preciselyspecified minimal requirements for a reasonablesocial welfare function.

Normative Analysis


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Normative Analysis

Minimal requirements for areasonable social welfare function

Unrestricted Domain: The social preference orderthat is derived from individual preferences must becom lete reflexive and transitive.

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Independence of irrelevant alternatives: Theassessment of two alternatives should beindependent of that of other alternatives.

Weak Pareto Principle

Nondictatorship: The preference of any one

individual may not be declared the social preference.

Normative Analysis


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Normative Analysis

Kenneth Arrow (1963) has shown that no socialdecision mechanism exists that can satisfy theseconditions.

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The result is sobering; it shows that no guarantee for

rational decisions exists in politics.

It also explains why collective decisions can be

arbitrary.

Pareto Efficiency


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Pareto Efficiency

Alternative 3: Pareto Criterion

Definition: Given an allocation x. A feasible allocation y -

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.

Definition: Given an allocation x. An feasible allocationyis weakly Pareto-better if all agents do not find yless preferable to x, and at least one agent prefers y.

Pareto Efficiency


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Pareto Efficiency

Definition (Pareto-efficiency): an allocation xisPareto-efficient if it does not allow any weak

Pareto-improvement.

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An allocation is Pareto-efficient if it isimpossible to make someone better off

without making someone worse off.

Pareto Efficiency


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Pareto Efficiency

Remarks:

This criterion gives every economic subject a vetoright.

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Efficiency has nothing to do with justice (whatever itsdefinition): An allocation in which an agent consumesall goods is Pareto-efficient.

In general there are many Pareto-efficient allocations.

Pareto Efficiency


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Pareto Efficiency

Example: Barter economy

2 agents, 2 goods x1 and x2

Initial allocation 1 and 1e e= =

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Utility functions

( ) ( )( )

1 1

1 11 12 11 12 2 21 22 21 22, und , ,, 0,1

u x x x x u x x x x

= =

and

Pareto Efficiency


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Pareto Efficiency

Calculation of Pareto-efficient allocations:

( ) 11 11 12 11 12,ax u x x x x

=

74

( )

, , ,

12 21 22 21 22 2

11 21

12 22

s.t. ,

1

1

u x x x x u

x x

x x

= =

+ =

+ =

Pareto Efficiency


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a y

Lagrange:

( ) ( )( )11

11 12 2 11 121 1 L x x u x x

=

75

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

1 11 1

11 12 11 12

11

11 12 11 1212

1

2 11 12

1 1 0

1 1 1 1 0

1 1 0

L x x x x

x

L

x x x xx

Lu x x

= =

= =

= =

Pareto Efficiency


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y

Marginal rates of substitution:

( ) ( )12 12

1 2

11 11

11 1 1

x x RS MRSx x

= = =

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Solve for x12 (contract curve):

( )

( )

( )11

1211

1

where1 1 1

x

x x

= =

Pareto Efficiency


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y

Edgeworth-Box: Contract curve

x12

x21 2xx

21

= e1

- xx11

x

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The contract curve is the set of all Pareto-efficientallocations.

x111 x22xx11

xx22 = e2- xx12xx12

~2~1

( )12

111 1x

x =

Pareto Efficiency


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y

How does a contract curve look like?

( )12

2

11 11

01 1

dx

dx x

= >

( ) ( )

( )

2

111242

11 11

2 1 1 11 1

xd xdx x

=

78

( )

( )

( )

( )( )

( )

2

122

11

2

12

2

11

2

12

2

11

1

If 1, then 0.1

1If 1, then 0.

1

1If 1, then 0.

1

d x

dx

d x

dx

d x

dx

= >

Pareto Efficiency


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y

The example demonstrates that an allocation

is only Pareto-efficient if the marginal rate ofsubstitution between two goods is identical

for both households.

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This statement can be extended to any

number of goods and households: An

allocation is only then Pareto-efficient if themarginal rate of substitution between anytwo goods is the same for all individuals.

Pareto Efficiency


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y

Edgeworth-Box: Inefficient allocation

x12x21 2xx21 = e1- xx11

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x111 x22xx11

xx22 = e2- xx12xx12

~2

~1

Pareto Efficiency


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Edgeworth-Box: Efficient allocation

x212xx21 = e1- xx11

81

x111 x22xx11

xx22 = e2- xx12xx12

~2

~1

Slopes of the indifference curves are

identical.

Pareto Efficiency


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Example: Production economy

Consider the following allocation problem: Player 1produces qB bananas and player 2, qO oranges.

82

Let the cost functions be c(qi) = qi, i= B,O.

Let the utility functions of both players for bananas

and oranges be u1(qO) and u2(qB) where

' ''0 und 0, 1, 2.i i

u u i> < =and

Pareto Efficiency


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Payoff to both players:

u1(qO) qB for player 1u2(qB) qO for player 2

83

Calculate the Pareto-efficient allocations:

( ) ( )

( )

1 2 1,

2 2

max

s.t.O B

O Bq q

B O

S S u q q

u q q S

=

=

Pareto Efficiency


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Lagrange function:

( )1 2 2L SB Bu u q q=

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Pareto-efficient production and consumptiongiven S2

[ ] ( )1 2' ' 1 0O Bu q u q =

( ) ( )* *2 2,B Oq S q S

Pareto Efficiency


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Curve of the Pareto-efficient allocations

( ){ ( )* *

1 1 2 2 2 2S S S SB Bu u q q

=

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envelope theorem):

The curve is concave:

11

2

S' 0

Su

=

Pareto Efficiency


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The curve of Pareto-efficient allocations:

S1Pareto-efficient

86

S2

Pareto-

inefficient

allocations

S1(S2)

Pareto Efficiency


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In order to get an initial idea of the relationship

between Walrasian equilibrium and Pareto-efficiency,

let us once again look at the first-order conditions of abarter economy with two households (h= 1,2) andtwo oods i= 1 2 .

87

( )

( )

11 12 21 22

1 11 12, , ,

2 2

2 21, 22 2

1 1

max ,

s.t. and

x x x x

h h

h h

u x x

x e u x x u= =

=

Pareto Efficiency


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Lagrange:

First order conditions:

( ) ( )( )

2 2 2

1 11 12 2 2 21 22 21 1 1, ,i hi hii h hL u x x e x u x x u = = =

= + +

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iMultipliers

( )

( )

1 11 12

1 1

2 21 22

2

2 2

, 0, 1,2

,0, 1,2

i

i i

i

i i

u x xL ix x

u x xLi

x x

= = =

= = =

Pareto Efficiency


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Identical marginal rates of substitution:

( ) ( )1 11 12 2 21 2211 1 21 1

1 2

, ,

,, ,

u x x u x xx x

MRS MRSu x x u x x

= = = =

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The multipliers can be interpreted as the

prices p1 and p2.

12 22x x

Pareto Efficiency


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The optimality conditions of the previous the

maximization problems are identical with the

optimality conditions of a Walrasianequilibrium!

90

In a Walrasian equilibrium, the followingholds:

1 1

1 22 2

und

p p

MRS MRSp p= =and

Pareto Efficiency


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The multipliers ireflect the fact that the endowment

of goods is scarce.

If we differentiate the Lagrange function with respectto the endowment ei we get i.

91

, i

could get if we had a bit more of good i. Since we have just seen that the multipliers and

prices enter in the same way into the first-order

conditions of the two problems, it is clear that prices

also reflect scarcity of goods.

Pareto Efficiency


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Edgeworth-Box: Walrasian equilibrium

x12x212xx21 = e1- xx11

92

x111 x22xx11

xx22 = e2- xx12xx12

~2

~1

x*

e*

Pareto Efficiency


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We have just seen that there is a close

relationship between Pareto efficiency andWalrasian equilibrium.

93

This relationship is investigated in thefollowing welfare theorems.

Normative Analysis


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94

The Welfare Theorems

First Welfare Theorem


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Theorem (First Welfare Theorem).

Consider a general equilibrium model with

local non-satiated references. Furthermore

95

let (p*, x*, y*) be a Walrasian equilibrium.Then the allocation (x*, y*) is Pareto-efficient.



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Proof (indirect):

Let (p*, x*, y*) be a Walrasian equilibrium and(x, y) an allocation that is Pareto-better.

96

-

that*

1 1

*

for household 1

2,...,h h

x x

x x h H =



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Why has household 1 not chosen x1 withprices p*?

Goods bundle x1 must be more expensive

97

than x1* with prices p*, otherwise household

1 would not have maximized its utility:

1 1* * *p x p x >



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For all other households,

must hold owing to local nonsatiation.

* * *h h

p x p x

98

Summing across all households yields:* * *

1 1

H H

h h

h h

p x p x= =

>



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Owing to local non-satiation, a utility-

maximizing consumption bundle must

exhaust the budget; i.e.,

H H H F

99

( )

( ) ( )

* * * * *

1 1 1 1

* * *

1 1 1

1

* * * * *

1 1 1 1

h h hf f

h h h f

H F H

h hf f

h f h

H F H F

h f h f

h f h f

p x p e p y

p e p y

p e p y p e p y

= = = =

= = =

=

= = = =

= +

= +

= + +



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The last inequality is an implication of thefirms profit maximization.

But this then gives

( )+>H F

h

H

h

H

h ypepxpxp*****

100

A contradiction of the assumption that theproposed allocation is feasible; that is

1 1 1

H H F

h h f

h h f

x e y= = =

+

+>

= ===

= ===

H

h

F

f

fh

H

h

h

H

h

h

h fhh

yepxpxp1 1

*

1

**

1

*

1 111

!



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"To say it in words"

An allocation B, which is Pareto-better

than competition allocation A, mustcost more than A, since, otherwise, the

latter would not represent the

101

competition equilibrium. In order to

afford B, household incomes would haveto rise. However, with the given

allocation this would only be possible

if profits were to rise. Since firms

behave in a profit-maximizing manner,Bs profit is lower than that of A.

Thus, B is not viable.



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The first welfare theorem shows that the marketmechanism achieves an efficient allocation.

The market mechanism is a very simple mechanism:

102

Economic agents only have to know prices and their

own preferences and production technology.

However, the question of who sets prices when all

economic agents are price takers remainsunanswered.



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Conditions for the first theorem to hold:

No external effectsThe utility of a consumer depends only on the

103

goo s un e t at e, mse , consumes.

Opposite example: cigarette consumption

Local nonsatiation.

Convexity is no condition.



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Definition (local nonsatiation): The property of local nonsatiation ofconsumer preferences states that for any bundle of goods there isalways another bundle of goods arbitrarily close that is preferred to it.

Notes (Definition and comments are from wikipedia):

1. Local nonsatiation is implied by monotonicity of preferences, but not viceversa. ence s a wea er con on.

2. There is no requirement that the preferred bundle y contain more of anygood - hence, some goods can be "bads" and preferences can be non-monotone.

3. It rules out the extreme case where all goods are "bads", since then the

point x = 0 would be a bliss point.

4. The consumption set must be either unbounded or open (in other words,it cannot be compact). If it were compact it would necessarily have abliss point, which local nonsatiation rules out.

Local Nonsatiation


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To illustrate that local nonsatiation is a

necessary condition for the first welfare

theorem, we consider an economy with twohouseholds and two goods in which this

105

characteristic is absent.

Local Nonsatiation


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Edgeworth-Box: Local nonsatiation

x12x21 2

106

x111 x22~2

x*

e

~2

~1

c n erence

curvex

Local Nonsatiation


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The allocation x* is a Walrasian equilibrium,but it is not Pareto-efficient!

107

Player 1 has local non-satiated preferences;i.e., he has thick indifference curves.

He is therefore indifferent to xand x*, but xisstrictly preferable for consumer 2!



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Theorem (Second Welfare Theorem):

Under certain conditions every Pareto-

108

Walrasian equilibrium by means of a suitableselection of property rights.



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Idea of the second welfare theorem

The Pareto-efficient allocation xshould be realized.Initial endowmentis e.

109

x

1

e


S 1 L k f i h h h b d i h


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Step 1: Look for price vector p, such that the budget constraint thatpasses through e, has the same slope as the tangent to the twoindifferent curves at point x.

2

110

1

e

x



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Step 2: Redistribution (by means of taxes and transfers) of theinitial allocation eto e'.

2

111

1

e'

x e


W di th lifi t i


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We now discuss the qualifier certain

conditions in the statement of the theorem.

112

Convexity

Local nonsatiation



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Convexity (Households)

113

.

Remarks: If a preference ordering is convex,then the upper contour set is convex.



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x2

{y: y x} Upper contour set ofx

114

x1

1

~1

x

Remark:{y: y x} is called the upper contour set of x.

{y: y> x} is called the strict upper contour set of x.


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Allocation x is Pareto-efficient, however notfeasible as a Walrasian equilibrium.

116

consumption bundle x2 = (x21, x22) on thebudget constraint line, consumer 1 wouldprefer the allocation y1 = (y11, y12).


Convexity (Firms)


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Convexity (Firms)

Every production set Yf is convex:

117

No increasing economies of scale

No indivisibility No fixed costs


Examples of non convex technologies:


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Examples of non-convex technologies:

Fixed costs IndivisibilityIncreasing

118

scale

0 00


Example of fixed costs


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a p s s

An economy is assumed to consist of a consumer, h= 1,and two goods n= 2, the work of the consumer and theconsumption goods produced by his labor. ("Robinson

"

119

- .

The production function is characterized by fixed costs.

Pareto-efficiency requires that the marginal rate of

substitution is equal to the technical rate of substitution(MRS = TRS).


Decentralization is possible


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Decentralization is possible

y2

~1

120y1 0

p

x

Yf


Decentralization is not possible


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Decentralization is not possible

y2~1

121

y1

p

x

Yf


Allocation x is Pareto-efficient but not


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Allocation xis Pareto efficient, but notachievable as a Walrasian equilibrium, since

it does not maximize the producers profit!

122

Problem: Technology Yf is not convex. Thefirm makes losses.

Interpretation of the Welfare Theorems

Until now we have discussed the welfare theorems from


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a theoretical point of view. On the next few slides

well point out some difficulties arising from adaptingthese findings to the real-world.

123

Our focus will be on the following issues:

1. Market Economy vs. Planned Economy

2. Redistribution of the initial allocation

3. Market Failure as a real-word problem

Market Economy vs. Planned Economy

From Wikipedia:


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p

The first theorem is often taken to be an analyticalconfirmation of Adam Smith's "invisible hand"

124

,toward the efficient allocation of resources.

The theorem supports a case for non-intervention inideal conditions: let the markets do the work and theoutcome will be Pareto efficient.



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Statement: The economic policy maker only has todetermine property rights in a market economy and canleave efficient allocation to the market. The marketeconomy is thus better than a planned economy.

125

This is a fallacious statement, as in a centrally plannedeconomy with perfect information exactly the samesolution can be achieved. Both systems are equivalent

in this respect.


What can be said if the assumption of complete


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informationby the planner is dropped?

- With a planned economy the likelihood of arriving at a

126

.thus almost always inefficient.

- With a market economy an efficient allocation isguaranteed (first welfare theorem). But it is very

improbable that the realized allocation coincides withthe targeted allocation.

Redistribution of the initial allocation

2. Redistribution of the initial allocation


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To reach the target allocation, the policy maker has

two fundamental ways to influence the initialallocation:

- -

127

Since reallocation occurs independently of thedemand behavior of economic subjects on the

markets, the marginal conditions remain intact.Thus, this type of reallocation is efficient.

Redistribution of the initial allocation

b) Commodity-/consumption-/income taxes


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These reallocations are dependent on the

transactions and the behavior of market subjects. Adifference arises between the purchase and sales

128

.

The second welfare theorem only holds for areallocation of type a. In the real world, however, a

reallocation of type b is almost always the caseobserved.

Market Failure as a real-word problem

3. Market Failure as a real-word problem


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The fundamental theorems indicate why markets might fail toprovide efficient allocations.

Reasons for market failures:

129

proper y r g s are no we e ne

incomplete markets

incomplete information

transaction costs

Non-convexities

Markets may also fail if the utility- and production functions ofthe agents are dependent on the consumption or production ofthe other economic subjects (external effects).


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130

Positive Analysis

Literature: Varian, H., Microeconomic AnalysisC


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(Chapter 17)

In this section we shall look at

131

Walrass Law,

Relative and absolute prices

Existence of Walrasian equilibrium.

Walrass Law

We consider a barter economy.


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A consumers demand is defined as

132

( )

( )

( )

1 ,

,

,

h h

h h

hN h

x p e

x p e

x p e

=

Walrass Law

The consumers excess demand (net


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demand) is

( ) ( ),h h h hz p x p e e=

133

Aggregate excess demand is then

( ) ( ) ( )( )1 1 ,

H H

h h h h

h hz p z p x p e e

= == =

Walrass Law


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Theorem (Walrass law). For everyprice vector p, thevalue of aggregate excess demand is equal to zero;

i.e.,

134

( ) 0 p z p =

Walrass Law

Proof


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Owing to monotonicity, every households demandsatisfies the budget constraint

135

Thus it follows that

Aggregate

( ) ( )1

0H

h

h

p z p p z p=

= =

,h h h

( ) ( )( ), 0h h h hp z p p x p e e = =

Walrass Law

Remarks:


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The value of excess demand is always zero.

136

The law implies that if N-1 markets are inequilibrium, the Nthmarket will also be inequilibrium.

Relative Prices

Demand is homogeneous of degree zero for

prices:


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prices:

*HomogenittsgradDegree of homogeneity

137

Interpretation: If one multiplies all prices by

the factor , demand does not change (nomoney illusion!).

( )

( ) ( )

*

0

, , ,h h h h h hx p e x p e x p e = =

Relative Prices

Cobb-Douglas example:


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For a barter economy with two goods we getthe following demand:

138

The budget constraint is

Demand for good 1 is homogeneous:

( ) ( )11 21 2

, and ,h h h h

x p e x p ep p

= =

1 1 1 2 2h hb p e p e= +

( )( ) ( )

( )1 1 2 2 1 1 2 21 11 1

, ,h h h h

h h h h

p e p e p e p e x p e x p e

p p

+ += = =

Relative Prices

An important implication is that only relative

prices are relevant to demand


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prices are relevant to demand.

139

freely.

The price of the first good is often given as

numraire...

Relative Prices

The prices then have to be changed using theformula


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1, 1,..,

ii

p p i N p= =

140

The price of good 1 is thus normalized to .

Every price is thus measured in the units of the firstgood.

1 1p =


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141

Walrasian Equilibrium

Existence

It is often important to know whether the


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It is often important to know whether the

objects that are being discussed actuallyexist.

142

If one assumes that certain things exist even

though they do not in fact exist, this can result

in funny conclusions.


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"Raelians" believe thatextraterrestials broughtlife to the each 25,000

143

years ago.

Existence

27.10.07 7:56 Kantonal police of the Valais

In the Valais paintball guns are used to shoot at


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In the Valais paintball guns are used to shoot at

Raelians.

144

. .being held by the Raelian sect on Friday evening in

the lower Valais village of Mige and shot aroundwith a paintball gun. Two people were slightly injured.

According to the Valais police report, approximately

40 members of the UFO-sect were conducting ameeting in a large hall in Mige when the unknownperson intruded on the crowd and fired. The culpritthen took flight.

Existence

Proposition (Humbug): The largest integer number is 1.


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Proof: Let us suppose that a finitely large integernumber exists and let it be called N. Then,

2

145

since Nis the largest number. Thus

But since Nis the largest number,

QED.

1N

1N=

Existence

We have only arrived at this nonsense

because we have assumed the existence of a


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because we have assumed the existence of a

largest integer number.

146

But there is no such number and therefore

the proof is humbug.

Existence

In order to investigate the existence of a Walrasian

equilibrium, we will assume that every good has a


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positive excess demand when its price is equal tozero; i.e.,

147

Then a Walrasian equilibrium is a price vector p*,such that aggregate excess demand is z(p*) = 0.

This means that supply = demand for every good.

or a ,..,i i p z p= =

Existence

Proof of existence

In order to prove its existence it must be shown that


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In order to prove its existence, it must be shown that

there is a price vector p*, such that z(p*) = 0; i.e.,

148

We thus have an equilibrium system with Nequationsand Nunknowns.

( )

1

0

0

0Nz p

=

=

=


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Existence

However, since z(p) is homogeneous of degree zeroin prices, we are free to select a price.


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Thus the system of equations which we would like to

150

-

N-1 unknown prices:

( )

( )

1

1

0

0

0

0N

z p

z p

=

=

=

=

Existence

A Walrasian equilibrium exists if we can find a

solution to this system of equations.


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151

of a fixed-point theorem.

The simplest fixed-point theorem is Brouwers

fixed-point theorem.

Existence

Digression: Brouwers Fixed-Point Theorem


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Theorem. If is a continuousfunction, then an xexists with f(x) = x.

11

:

NN

SSf

152

Proof (for N= 2): Consider a continuous function .

The theorem states that this function has a

fixpoint; i.e., such that

:[0,1] [0,1]f

[ ]0,1x ( ) . f x x=

Existence

Define the function .

The function gmeasures the distance between f(x)

( ) ( )g x f x x=


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and the diagonal:f(x)

1530 1

1

x

Existence

A fixpoint x* of the function fthus satisfies

g(x*) = 0. f(x)


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Since f(0) [0,1], it holds that1

154

- .

Since f(1) [0,1], it holds that

g(1) = f(1)-1 0.

Since the function fis continuous, it follows at anx* [0,1] exists, such that g(x*) = 0 = f(x*) - x*.End of digression.

0 1x

Existence

Proof of existence (continuation):

We allow that excess demand z(p*) can be negative


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We allow that excess demand z(p ) can be negative

in a Walrasian equilibrium (Demand < Supply).

155

A Walrasian equilibrium is then a price vector p*,

such that aggregate demand z(p*) 0.

The proof of existence involves to show that a price

vector p*exists, such that z(p*) 0.


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Existence

Unity simplex of

N

N

+


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1

11

N N

i

iS p p

+

=

= =

157

S1S2

2p

2p

1p

1

p

3p

Existence

The proof of existence involves to show thata normalized price vector p SN-1 exists such


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that z(p) 0.

158

The proof uses Brouwers fixed-point

theorem.

Existence

Theorem (Existence): If excess demand

is continuous, then a price1: N Nz S


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, p

vector p* SN-1 exists such that z(p*) 0.

:z S

159

Proof:

Consider the function 11: NN SSg

( )( )

( )( )( )

1

max 0,1,...,

1 max 0,

i ii N

j

j

p z pg p i N

z p=

+

= =+

Existence

Since z(p) and max(...) are continuousfunctions, then g(p) is also continuous.


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Since the ran e of this function is1N

=

160

the simplex SN-1.

Interpretation of g: If there is a surplusdemand for good i; i.e., z

i(p) > 0, then the

price of this good will rise.

1i=

Existence

The function gthus has the properties that weneed in order to apply Brouwers fixed-point


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theorem.

161

It follows from Brouwers fixed-point theorem

that a p* exists such that p* = g(p*); and

( )( )

( )( )

*

*

1

max 0, *(*) 1

1 max 0, *

i i

i N

j

j

p z p p i ,...,N

z p=

+= =

+

Existence

It still needs to be proved that the fixpoint p*is a Walrasian equilibrium; i.e., z(p*) 0.


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*

162

( )( ) ( )( )*1

max 0, * max 0, * 1N

i j i

j

p z p z p i ,...,N =

= =

Existence

Multiplication of the equation with zi(p*)

( ) ( )( ) ( ) ( )( ) ,...,Nipzpzpzppz iiN

jii 1*,0max**,0max**

==


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j 1

=

163

,

Walrass law thus implies

( ) ( )( ) ( ) ( )( )*1 1 1

0

* max 0, * * max 0, *

N N N

i i j i i

i j i

z p p z p z p z p= = =

=

=

( ) ( )( )1

0 * max 0, *N

i i

i

z p z p=

=

Existence

Each term

( ) ( )( )* max 0, * 1,...,i i z p z p i N =


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is greater than or equal to zero:

( ) ( )( )

164

( ) ( ) ( )( )

( ) ( ) ( )( ) ( )2

If * 0, then * max 0, * 0

If * 0, then * max 0, * *

i i i

i i i i

z p z p z p

z p z p z p z p

=

> =

Existence

If one term were indeed greater than zero,

( ) ( )( )0 * max 0, *N

i i z p z p=


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could not be satisfied.1i=

165

Thus it holds that zi(p*) 0 i= 1,..,N.

Therefore p* is a Walrasian equilibrium.

QED

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03 Micro Walras