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1
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.
16
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 + + +
17
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
18
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.
19
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
22
Macroeconomics
Finance
Trade theory
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Introduction Historical description of general equilibrium
theory in a nutshell
23
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.
24
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.
25
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.
26
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.
29
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.
30
Goods are perfectly divisible.
There is a market for every good (completemarkets).
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The ModelPrices
Let pjbe the price of goodj.
31
We then call a pricevector.
( )1,....,N
N p p p
+=
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The ModelProducers (firms)
There are f= 1,, F producers.
32
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
33
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
35
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.
36
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:
37
( )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=
39
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
41
.
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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 =
42
( )
( )
*
* * * *
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.
43
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
44
right?)
Positive: Existence and characteristics
Empirical: Which real markets fit this model?
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The ModelQuestions
Normative: Welfare properties
45
Positive: Existence and comparative statics
Empirical: does the model fit the data
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Overview
Behavioralhyothesis
Exogenousdata
Endogenousdata
46
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
47
1 2
households.
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The ModelExample: Barter economy
Initial endowment
1 0 and 0 1e e e e e e= = = =
48
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
49
x111 x22ee11
ee22 = e2- ee12ee12
~2
~1
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The ModelCalculation of individual demand functions
Household 1 solves the following problem:
50
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 = +
51
( )
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
= =
52
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
+=
53
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+ = +
54
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 = =
55
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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56
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.
57
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?
58
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:
59
= 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
60
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.
61
.
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)?
62
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.
63
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.
64
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)
66
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
67
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.
68
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.
69
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 -
70
.
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.
71
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.
72
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= =
73
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
= = =
76
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
77
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
= >
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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.
79
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
80
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
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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
=
=
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Lagrange function:
( )1 2 2L SB Bu u q q=
84
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
=
85
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 = = =
= + +
88
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
= = = =
89
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.
First Welfare Theorem
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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 =
First Welfare Theorem
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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 >
First Welfare Theorem
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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= =
>
First Welfare Theorem
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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
= = = =
= = =
=
= = = =
= +
= +
= + +
First Welfare Theorem
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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
!
First Welfare Theorem
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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.
First Welfare Theorem
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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.
First Welfare Theorem
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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.
First Welfare Theorem
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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!
Second Welfare Theorem
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Theorem (Second Welfare Theorem):
Under certain conditions every Pareto-
108
Walrasian equilibrium by means of a suitableselection of property rights.
Second Welfare Theorem
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Idea of the second welfare theorem
The Pareto-efficient allocation xshould be realized.Initial endowmentis e.
109
x
1
e
Second Welfare Theorem
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
Second Welfare Theorem
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Step 2: Redistribution (by means of taxes and transfers) of theinitial allocation eto e'.
2
111
1
e'
x e
Second Welfare Theorem
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
Second Welfare Theorem
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Convexity (Households)
113
.
Remarks: If a preference ordering is convex,then the upper contour set is convex.
Second Welfare Theorem
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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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Second Welfare Theorem
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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).
Second Welfare Theorem
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
Second Welfare Theorem
Examples of non convex technologies:
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Examples of non-convex technologies:
Fixed costs IndivisibilityIncreasing
118
scale
0 00
Second Welfare Theorem
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).
Second Welfare Theorem
Decentralization is possible
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Decentralization is possible
y2
~1
120y1 0
p
x
Yf
Second Welfare Theorem
Decentralization is not possible
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Decentralization is not possible
y2~1
121
y1
p
x
Yf
Second Welfare Theorem
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.
Market Economy vs. Planned Economy
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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.
Market Economy vs. Planned Economy
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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