Cracow University of Economics
Introduction
to Mathematical Economics
General Equilibrium Theory
and Welfare Economics
by
Andrzej Malawski
Cracow 2012
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Contents
1. Introduction ............................................................................... 3
1.1. Historical remarks ....................................................... 3
General description of the economy ..................................... 4
2. Commodity/price space Rl ......................................................... 5
3. Production system ...................................................................... 6
3.1. General description ...................................................... 6
3.2. Assumptions on production sets ................................... 7
3.3. Profit maximization ...................................................... 8
4. Consumption system ................................................................... 9
4.1. General description ....................................................... 9
4.2. Consumption sets ......................................................... 10
4.3. Preferences .................................................................. 11
4.4. Preference maximization .............................................. 12
5. Debreu economy with the private ownership ............................... 13
6. General competitive equilibrium .................................................. 14
7. Pareto optimality ......................................................................... 15
7.1. Social preference relation and Pareto optimum ............. 15
7.2. Fundamental theorems of welfare economics .................16
8. Mathematical appendix ................................................................ 17
8.1. Sets and relations .......................................................... 17
8.2. Continuity ..................................................................... 18
8.3. Fixed points .................................................................. 19
8.4. Exercises …………………………………………….. … 19
References .................................................................................. 23
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1. Introduction
1.1. Historical remarks
The history of general equilibrium theory and welfare economics as a part of
mathematical economics can be divided after the Handbook of Mathematical Economics
(1982) into three broad and overlapping periods:
- the calculus-based marginalist period (1838 - 1947),
- the set-theoretic / linear models period (1948 -1960),
- the current period of integration (1961 - present).
In the first period economics borrowed methodologies from the physical sciences, so
that the basic mathematical tool was the calculus, particularly the use of total and partial
derivatives and Lagrange multipliers to characterize maxima. The mathematical foundations
of the modern theories of the consumer, the producer, oligopoly, and general equilibrium were
developed in this period. A seminal work of Cournot (1838) is assumed to be the starting
point of mathematical economics, where the problems of the interaction of firms and
consumers in single markets, in the cases of perfect competition and of monopoly, were
rigorously studied.
In the set-theoretic / linear models period the earlier calculus basis for mathematical
economics was replaced by a set-theoretic set-up and by linear models. It meant greater
generality, because the classical assumptions of smooth functions could be dropped and
replaced by more general relations.
The basic mathematical tools, such as set-theoretic
approach, functional analysis, convexity and elements of topology, were applied to the
analysis of economic phenomena. The main ideas and results of this period culminated in the
book of Debreu (1959) on the theory of general economic equilibrium, where the state of the
theory was summerized and the new extensions developed. This classic and extremely
influential book is still a standard approach and the starting point of many present
developments. The book in question is also the basis for our elaboration.
In the current period of integration modern mathematical economics combines the
formal techniques of the previous periods, i.e., elements of calculus, set theory and linear
models. It is also a period in which mathematical concepts and methods have been extended
to virtually all areas of economic science. Some of them can be mentioned: global analysis,
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duality theory, aggregate demand functions, core of economy and markets with a continuum
of traders, temporary equilibrium, computation of equilibrium prices, social choice theory,
optimal taxation, optimal growth theory, organization theory and others.
1.2. General description of the economy
The general model of a social system that will be here presented includes as a special
case the following economy (economic system). The agents of the economy produce, trade,
and consume l commodities. Each one of these commodities is a good or a service with
specified physical characteristics available at specified date and location. The different types
of human labour are among the commodities.
We distinguish two kinds of agents, namely consumers and producers. They role is to
select and carry out an action, i.e., a consumption or production plan, respectively, which is a
vector in the commodity space Rl. The economic agents are characterized by the limitations
on their choice, and by the choice criterion. The production plan of a producer is constrained
to belong to a given set representing his limited technological knowledge. Under those
constraints he tries to maximize his profit. On the other hand, the choice limitations of a
consumer are of two kinds: firstly, the consumption plan must satisfy certain constraints of a
psycho-physiological nature; secondly, the consumer is limited in his decision by the budget
constraints. Under those constraints he tries to maximize his preferences. All the actions of
economic agents are represented by some points of Rl.
It should be mentioned also, that the economic agents in question are price takers, i.e.,
they are not allowed to influence the prices of commodities which are interpreted as the states
of an environment. The price system to prevail is also represented by a point of Rl.
Finally, to complete the description of the whole economic system, we define its total
resources as a vector ω in the commodity space Rl, so that consumers own the resources.
They are also shareholders of the producers’ profits.
In summary, the economic system E is described by the following formal scheme:
E = (Rl, P, C, θab, ω)
where
Rl is the commodity/price space,
P is the production system,
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C is the consumption system,
θab are the shares of the consumers in the producers profits,
ω is the the total resources vector.
All the categories listed above will be explained in some detail in the next chapters.
2. Commodity/price space Rl
Commodities can be divided into goods and services. Each commodity is completely
specified by its physical characteristics, its location, and date at which it is available. The
traditional theory usually assumes that there exists a finite number
l of
commodities
implying that a finite specification of physical characteristics, location, etc. suffices for the
problems studied. Quantities of each commodity are measured by real numbers. A commodity
bundle, i.e. a list of real numbers (xh), h = 1,..., l, indicating the quantity of each commodity,
can be described therefore as an l-dimensional vector x = (x1,..., xl) and as a point in ldimensional Euclidean space Rl, the commodity space. Under perfect divisibility of all
commodities any real number is possible as a quantity for each commodity so that any point of
the commodity space Rl is a possible commodity bundle.
To justify negative real numbers, the following convention is introduced. Each
commodity bundle is assumed to be an input-otput vector in Rl, where what is made available
to an economic agent is called an input for him, and what is made available by an economic
agent is called an output for him. The economic agents are divided into two groups: producers
and consumers. Consequently, for producers inputs are represented by non-positive numbers
and outputs by non-negative numbers. For consumers the reverse convention is assumed, i.e.,
inputs are non-negative and outputs non-positive.
The price ph of a commodity h = 1,...,l is a real number which is the amount paid in
exchange for one unit of the commodity. With the specification of location and date the theory
assumes the general convention that the prices of all commodities are those quoted now on the
floor of the exchange for delivery at different locations and at different dates. A price system
or a price vector p = (p1,..., pl) can thus be represented by a point in Euclidean space Rl. The
value of a commodity bundle given a price vector p is ∑h = 1l phxh = px.
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3. Production system
3.1. General description
The production of the bth producer (b ∈ B) is also an l-list of the quantities of the
various commodities that he consumes and that he produces, but we represent his inputs by
negative numbers and his outputs by positive numbers. Thus, his production is a point yb in
the commodity space Rl, and his technological knowledge defines the non-empty subset Yb of
Rl of his possible productions. Given a price vector p in Rl, the bth producer strives to choose
in his production set Yb a production yb that maximizes his profit pyb.
So, the production system is represented by a two-range relational system:
P = (B, Rl; y, p, η, π),
where:
B = {b1, … , bn} is a finite set of the producers,
Rl is an l-dimensional commodity/price space,
y ⊂ B×P0(Rl) is a correspondence of production sets which to every producer
b ∈ B assigns a production set Yb ⊂ Rl being a subset of the commodity space and
representing the producer’s feasible production technology,
p ∈ Rl is a price system,
η ⊂ B×P0(Rl) is a correspondence of supply which to every producer b ∈ B assigns a
set η(b) of the production plans maximizing his profit pyb in a price system p, it is to say:
η( b ): = ηb ( p ): = { y b ' ∈Yb : py b ' = max y
b ∈Yb
py b },
π: B → R is a maximal profit function which measures the maximum profit value in
the set of plans η(b) , i.e., for every b ∈ B:
π ( b ): = π b ( p ): = max y ∈η( b ) py b .
Given a production yb ∈ Yb for each producer, y = ∑ b ∈ B yb is called the total
production. The set Y = ∑ b ∈ B Yb is called the total production set.
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3.2. Assumptions on production sets
The following assumptions on production sets will be discussed within the model
under consideration.
1. Yb is closed (continuity),
i.e., let (ybq) be a sequence of production plans; if all the ybq are possible for the bth
producer, and if ybq → yb0, then yb0 is also possible for the bth producer.
1’. Y is closed.
2. 0 ∈ Yb (possibility of inaction),
i.e., the bth producer has the possibility of doing nothing.
2’. 0 ∈ Y.
3. Y ∩ Rl+ ⊂ {0} (impossibility of free procuction),
i.e., a possible total production whose inputs are all null has all its outputs null.
4. Y ∩ (-Y) ⊂ {0} (irreversibility),
i.e., if the total production y, whose inputs and outputs are not all null, is possible, then the
total production -y is not possible. The productive process can not be reversed since, in
particular, production takes time and commodities are dated.
5. Returns to scale.
Given Yb, one says that:
5.1. non-decreasing returns to scale prevail if for any possible yb one can arbitrarily
increase the scale of operations, i.e., yb ∈ Yb ⇒ tyb ∈ Yb for t > 1,
5.2. non-increasing returns to scale prevail if for any possible yb one can arbitrarily
decrease the scale of operations, i.e., yb ∈ Yb ⇒ tyb ∈ Yb for 0 ≤ t < 1,
5.3. constant returns to scale prevail if for any possible yb one can arbitrarily change
the scale of operations, i.e., yb ∈ Yb ⇒ tyb ∈ Yb for t ≥ 0,
6. Yb + Yb ⊂ Yb (additivity),
i.e., if yb1 and yb2 are productions possible for the bth producer, so is yb1 + yb2.
7. Yb is convex (convexity),
i.e., if yb1 and yb2 are productions possible for the bth producer, so is their weighted average
tyb1 + (1 - t)yb2, with arbitrary positive weights .
7’. Y is convex.
8. Yj is a cone with vertex 0 (constant returns to scale).
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9. (-Rl+) ⊂ Y (free disposal),
i.e., if a total production plan has all its outputs null, it is possible. In other words, it is
possible for all producers together to dispose of all commodities.
10. (-Rl+ + Y) ⊂ Y,
i.e., if a total production plan is possible, so is one where no output is larger and no input
smaller (in absolute value). In other words, only a „worse” production is possible.
11. Yj is compact (a closed and bounded set).
3.3. Profit maximization
Given a price system p and a production yb, the profit of the bth producer is defined
by pyb. The total profit is py = p(y1 + ... + yn).
Given the price system p, the bth producer chooses his production in his production
set Yb so as to maximize his profit. The resulting action is called an equilibrium production of
the bth producer relative to p.
When p ≠ 0 one has the following geometric situation. If yb a maximizer, the set Yb
is contained in the closed half-space below the hyperplane H through yb with normal p. The
set of profit maximizers is the intersection of Yb and H.
However, it must keep in mind that given an arbitrary p the set of profit maximizers
may be empty. Thus, we define the set Tb = {p ∈ Rl; pyb has a maximum for yb ∈ Yb}. This
set is the domain of the supply correspondence of bth producer η(b) ⊂ Tb x Yb which is
defined as follows:
η( b ): = ηb ( p ): = { y b ' ∈Yb : py b ' = max y
b ∈Yb
py b }, .
Similarly, the set Tb is the domain of the profit function π(b): Tb → R of bth
producer such that π ( b ): = π b ( p ): = max y ∈η( b ) py b .
The total supply correspondence is defined by η(b) = ∑b ∈B ηb(p), and the total profit
function as π(b) = ∑b ∈B πb(p).
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4. Consumption system
4.1. General description
The consumption of the ath consumer (a ∈ A, card A = m) is an l-list of the quantities
of the various commodities that he consumes or that he produces (the latter being typically
human labour). His inputs are represented by positive numbers and his outputs by negative
numbers. Thus, his consumption is a point xa in the commodity space Rl. However, his
consumption cannot be chosen arbitrarily in Rl, because of the psycho-physical constraint of
this individual. We denote by Xa , a non-empty subset of Rl, the set of the possible
consumptions of the ath consumer. The tastes of this consumer are described by a complete,
reflexive, transitive binary preference relation
a
on his consumption set Xa and x
a
x’ is
read as „x’ is at least as desired by the ath consumer as x”.
Given a price-vector p in Rl listing the prices of all the commodities, the net value of
the consumption xa of the ath consumer is the inner product pxa. It must be at most equal to
his wealth wa. Under this constraint the ath consumer strives to satisfy his preferences
he strives to choose a greatest element for the relation
a
a,
i.e.
in the set {x ∈ Xa; pxa ≤ wa}. To
complete the description of the characteristics of the ath consumer, and to explain how his
wealth is formed, we specify his initial endowment of each one of the commodities as a vector
ea in the commodity space Rl, and his shares of the profits of the producers, θab being the
share of the profit of the bth producer owned by the ath consumer, where a ∈ A and b ∈ B.
The numbers θab are positive or zero, and for every b, ∑a ∈ A θab = 1. Thus, if the profit of the
bth producer is rb, the wealth of the ath consumer is wa = pea + ∑b ∈ B θabrb.
So, the consumption system is represented by a three-range relational system
C = (A, Rl, P; x, e, ε, p, β, ϕ),
where
A = {a1, …, am} is a finite set of the consumers,
Rl is an l-dimensional commodity/price space,
P ⊂ P(R2l) is the family of all preference relations defined on the commodity space
Rl,
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x ⊂ A×P0(Rl)
is a correspondence of consumption sets which to every consumer a
∈ A assigns a consumption set Xa = x(a) being a subset of the commodity space Rl and
representing the consumer’s feasible consumption plans,
e ⊂ A×Rl is an initial endowment mapping which to every consumer a ∈ A assigns
some initial endowment vector e(a) ∈ x(a),
ε ⊂ A×P(R2l)
preference relation
is a correspondence which to every consumer a ∈ A assigns a
a
∈ P, restricted to the consumption set x(a),
p ∈ Rl is a price system,
β ⊂ A×P0(Rl) is a correspondence of budget sets which to every consumer
a ∈ A assigns their set of budget constraints β(a) ⊂ x(a) with the price system p and the
initial endowment e(a), i.e., for every a ∈ A:
β (a ): = β ( p ,e ( a )) (a ): = {x ∈ x (a ): px ≤ pe(a )},
ϕ ⊂ A×P0(Rl)
is a demand correspondence which to every consumer a ∈ A assigns
their consumption plans maximizing their preferences on the budget set β(a), i.e., for every
a ∈ A:
ϕ(a) := ϕ(ε(a),p,e(a))(a) := {x ∈ β(p,e(a))(a): ∀ x’ ∈ β(p,e(a))(a): x’
a
x}.
4.2. Consumption sets
The following assumptions on consumption sets will be discussed within the model
under consideration.
1. Xa is closed (continuity).
1’. X is closed.
2. Xa has a lower bound for ≤ (lower boundedness),
i.e., there is a point χa in Rl such that χa ≤ xa for all xa in Xa, what means that Xa ⊂ {χa} +
Rl+.
2’. Xa has a lower bound for ≤.
3. Xa is connected (connectedness).
This means, loosely speaking, that Xa is made of one piece.
4. Xa is convex (convexity).
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4.3. Preferences
Among alternative commodity bundles in the consumption set Xa, the consumer a is
assumed to have preferences represented by a binary relation
and x2 in Xa the statement x1
a
a
on Xa. For two bundles x1
x2 is read as „x2 is at least as good as x1”. Three basic
axioms are usually imposed on the preference relation which are often taken as a definition of
a rational consumer.
Axiom 1 (Reflexivity).
For all x ∈ Xa, x
a
x, i.e. any bundle is as good as itself.
Axiom 2 (Transitivity).
For any three bundles x1, x2, x3 in Xa, x1
a
x2 and x2
a
x2 or x2
a
x3 implies x1
a
x3.
Axiom 3 (Completness).
For any two bundles x1 and x2 in Xa, x1
A preference relation
a
a
x1.
which satisfies these three axioms is a complete preordering
on Xa and will be called a preference order. Two other relations can be derived immediately
from a preference order. These are the relation of strict preference <<a and the relation of
indifference ~a.
Definition 4.1. A bundle x2 is said to be strictly preferred to a bundle x1, i.e. x1 ≺a x2 if and
only if x1
a
x2 and not x2
a
x1.
Definition 4.2. A bundle x1 is said to be indifferent to a bundle x2, i.e. x1 ~a x2 if and only if
x1
a
x2 and x2
With
a
a
xa.
being reflexive and transitive the strict preference relation is clearly
irreflexive and transitive. The indifference relation ~a defines an equivalence relation on Xa,
i.e. ~a is reflexive, symmetric, and transitive. Axioms 1-3 describe order properties of a
preference relation which have intuitive meaning in the context of the theory of choice.. This
is much less so with the topological conditions which are usually assumed as well. The most
common one is given in Axiom 4 below.
Axiom 4 (Continuity)
For every x’ ∈ Xa the sets {x ∈ Xa; x
The set {x ∈ Xa; x’
a
a
x’} and {x ∈ Xa; x’
a
x} are closed relative to Xa.
x} is called the upper contour set and {x ∈ Xa; x
a
x’} is
called the lower contour. Intuitively Axiom 4 requires that the consumer behaves consistently
in the „small”, i.e. given any sequence of bundles xn converging to a bundle x such that for
all n each xn is at least good as some bundle x’, then x is also at least as good as x’.
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For a preference order, i.e. for a relation satisfying Axioms 1-3, the intersection of the
upper and lower contour sets for a given point x defines the indifference class Ia(x) :=
= {x’ ∈ Xa; x’ ~a x} which is a closed set under Axiom 4. For alternative bundles x these are
the familiar indifference curves for the case of Xa ⊂ R2. Axioms 1-4 together also imply that
the upper and the lower contour sets of the derived strict preference relation ≺a are open, i.e.
{x ∈ X; x ≺a x’} and {x ∈ Xa; x’ ≺a x} are open.
4.4. Preference maximization
Given the price system p and a real number wa, the wealth of athe consumer, he
chooses his consumption xa ∈ Xa so that his expenditure pxa ≤ wa. The point w = (wa) of
Rm is called the wealth distribution. The point (p, w) of Rl+m is called the price-wealth pair.
The set Sa is defined by Sa = {(p, w) ∈ Rl+m; there is xa ∈ Xa such that pxa ≤ wa}.
The budget set correspondence βa ⊂ Sa x Xa is defined by βa(p, w) = {xa ∈ Xa; pxa ≤ wa}.
For wa = pe(a) this definition is compatible with the previous formulation in 4.1:
β (a ): = β ( p ,e ( a )) (a ): = {x ∈ x (a ): px ≤ pe(a )}, .
Given a price-wealth pair (p, w) in Sa, the ath consumer chooses in the set βa(p, w)
a greatest element for his preference preordering
a.
The resulting action is called his
equilibrium consumption relative to (p, w).
The set Sa’ is defined by Sa’ = {(p, w) ∈ Rl+m; βa(p, w) has a greatest element for
a}.
The demand correspondence of ath consumer ϕa ⊂ Sa’ x Xa is defined by
ϕa(p, w) = {xa ∈ βa(p, w); xa is a greatest element of βa(p, w) for
a}.
The total demand correspondence ϕ ⊂ ∩ a ∈ A Sa’ x X is defined by
ϕ(p, w) = ∑ a ∈ A ϕa(p, w). For wa = pe(a) this definition is compatible with the previous
formulation in 4.1:
ϕ(a) := ϕ(ε(a),p,e(a))(a) := {x ∈ β(p,e(a))(a): ∀ x’ ∈ β(p,e(a))(a): x’
a
x}.
5. Debreu economy with the private ownership
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A private ownership economy Ep is such a combination of a production system P and a
consumption system C that the consumers share in the producers’ profits (the shares are
measured by a mapping θ) and some fixed (initial) total resource ω of the economy Ep is
the consumers’ property. Thus the private ownership economy E can shortly be described in
the form (cf. part 1):
E = ( Rl, P, C, θ, ω),
where
P is a production system,
C is a consumption system,
θ ⊂ (A×B)×R+
is a function describing the consumers’ shares in the producers’
profits, i.e., for every (a,b) ∈ A×B the number θab := θ(a,b) ∈ [0,1] measures the consumer
a’s share in the producer b’s profit and there is, for every b ∈ B,
∑
a ∈A
θ ab = 1,
ω: = ∑a∈A e ( a ), so that ω ∈ Rl.
Now the formula:
E = (B, A, Rl, P; y, p, η, π, x, e, ε, β, ϕ, θ, ω)
describes a Debreu private ownership economy as a four-range relational system (henceforth:
a (Debreu) economic system).
The action of a Debreu economic system E can be summerized in the following logical
order.
1) the price system is p,
2) the bth producer tries to maximize his profit on Yb; yb does it,
3) the profit πb(p) = pyb is distributed to shareholders,
4) thus the wealth of the ath consumer is wa = pea + ∑b ∈ B θab πb(p),
5) this consumer tries to satisfy his preferences on Xa subject to his wealth constraint;
xa does it,
6) if the actions xa, yb satisfy the market equilibrium equality:
x - y = ∑a ∈ A xa - ∑b ∈ B yb = ω,
the economy is in equilibrium, i.e., every agent, given the price system and the actions ot the
other agents, has no incentive to choose a different action, and the state of the economy is a
market equilibrium
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6. General competitive equilibrium
A state of the economy E is an m-list (xa) of the consumptions of the various consumers, an
n-list of the productions of the various producers, and a price vector p. The state ((xa*), (yb*),
p*)) is an equilibrium if
(a) for every a, xa* is a best (maximal) element for
a
of {x∈ Xa; p*x ≤ p*ea +
∑b ∈ B θab p*yb*},
(b) for every b, yb* maximizes p*yb on Yb,
(c) ∑a ∈ A xa* - ∑b ∈ B yb* - ∑a ∈ A ea = 0.
Condition (a) says that every consumer has chosen in his consumption set a
consumption that satisfies his preferences best under his budget constraint. Condition (b) says
that every producer has maximized his profit in his production set. Condition (c) says that for
every commodity the excess of net demand over total resources is zero. The equilibrium
defined by conditions (a), (b), and (c) is competitive in the sense that every agent behaves as if
he had no influence on prices and considers them as given when choosing his own action.
In this context, the fundamental role plays the theorem on the existence of an
equilibrium.
Theorem 6.1. The economic system E has an equilibrium if
for every a,
Xa is closed, convex, and has a lower bound for ≤,
there is no satiation consumption in Xa,
for every x’ ∈ Xa the sets {x ∈ Xa; x
a
x’} and {x ∈ Xa; x’
a
x} are closed relative
to Xa,
if x and x’ are two points of Xa and t is a real number in (0, 1), then x ≺a x’ implies
x ≺a (1 - t)x + tx’,
there is xa0 in Xa such that xa0 << ea;
for every b,
0 ∈ Y b,
Y is closed and convex,
Y ∩ (-Y) ⊂ {0},
(- Rl+) ⊂ Y.
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The standard proof of the theorem 6.1 is based on the Kakutani theorem.
7. Pareto optimality
7.1. Social preference relation and Pareto optimum
Consider the economic system E = (Rl, P, C, θab, ω). Let be given two attainable
states of E, ((xa), (yb)) and ((xa’), (yb’)). The second is said to be at least as desired as the first,
and one writes ((xa), (yb))
((xa’), (yb’)), if, for every a, xa
It is easy to check that the social preference relation
a
xa’ .
, defined on the set M of
attainable states of E is reflexive and transitive, but it may not be complete, i.e., two atainable
states of E may not be comparable.
Moreover, the two states ((xa), (yb)) and ((xa’), (yb’)) are said to be indifferent if for
every consumer, his consumption in the two states are indifferent, i.e.,
((xa), (yb)) ~ ((xa’), (yb’)) means: for every a, xa ~a xa’
An optimum of the economy E can be now defined as an attainable state to which no
attainable state is socially preffered. In such a state a better satisfaction of the preferences of a
consumer necessarily occurs at the expense of the satisfaction of the preferences of another.
So, we can assume the following definition.
Definition 7.1. An optimum of E is a maximal element of M for
.
The theorem below gives the sufficient conditions for the existence of an optimum.
Theorem 7.1. The economic system E has an optimum if:
for every a,
Xa is closed, connected, and has a lower bound for ≤,
for every x’ ∈ Xa the sets {x ∈ Xa; x
a
x’} and {x ∈ Xa; x’
to Xa,
Y is closed, convex, and satisfies Y ∩ (Rl+) = {0},
ω ∈ X - Y.
a
x} are closed relative
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7.2. Fundamental theorems of welfare economics
To compare the categories of general equilibrium and Pareto optimum, the concept of
an equilibrium relative to the price system p is introduced.
Definition 7.2. A state ((xa*), (yb*)) of E is an equilibrium relative to the price
system p in Rl if:
(1) xa* is a greatest element of {xa ∈ Xa; pxa ≤ pxa*} for
a,
for every a ∈ A,
(2) yb* maximizes pyb on Yb, for every b ∈ B,
(3) x* - y* = ω.
Theorem 7.2. (the first theorem of welfare economics): an equilibrium relative to a
price system is an optimum.
Let E be an economy such that, for every a,
(1) Xa is convex,
(2) if x and x’ are two points of Xa such that x ≺a x’, and t is a real number in
(0, 1), then x ≺a (1 - t)x + tx’.
An equilibrium ((xa*), (yb*)) relative to a price system p , where no xa* is a satiation
consumption, is an optimum.
Theorem 7.3. (the second theorem of welfare economics): an optimum is an
equilibrium relative to a price system.
Let E be an economy such that, for every a,
(1) Xa is convex,
(2) for every xa’ in Xa, the sets {x ∈ Xa; x
a
x’ } and {x ∈ Xa; x’
a
x } are closed in
Xa,
(3) if x and x’ are two points of Xa such that x ≺a x’, and r is a real number in
(0, 1), then x ≺a (1 - t)x + tx’,
(4) Y is convex.
Given an optimum ((xa*), (yb*)) where some xa’* is not a satiation consumption, there
is a price system p ≠ 0 such that:
(α) xa* minimizes pxa on {xa ∈ Xa; xa*
a
(β) yb* maximizes pyb on Yb, for every b.
xa}, for every a,
17
8. Mathematical appendix
8.1. Sets and relations
Definition 8.1. If A and B are two sets, their product A x B is the set
{(x, y); x ∈ A, y ∈ B}.
If A = B = R, the set of real numbers, then their product R x R, denoted R2, is the set
of all ordered pairs of real numbers.
Definition 8.2. If A and B are two sets, their algebraic sum A + B is the set
{x + y; x ∈ A, y ∈ B}.
Definition 8.3. If A and B are two sets, a relation between A and B is a subset of A x B.
If a relation is a subset of A x A, we call it a relation on A.
Relations are often used to describe relationships between objects.
Definition 8.4. Let S be a relation on the set A. The relation S is said to be
a) reflexive if ∀ a ∈ A a S a,
b) symmetric if ∀ a, b ∈ A a S b ⇒ b S a,
c) transitive if ∀ a, b, c ∈ A a S b ∧ b S a ⇒ a S c,
d) antisymmetric if ∀ a, b ∈ A a S b ∧ b S a ⇒ a = b,
e) complete if ∀ a, b ∈ A a S b ∨ b S a.
An equivalence relation is any relation which satisfies a), b), and c).
A preorder is any relation which satisfies a) and c).
A partial order is any relation which satisfies a), c), and d).
Definition 8.5. Given two sets A and B, f is said to be a function or mapping from A into
B if for each x ∈ A there exists unique y ∈ B such that y = f(x). The set A is called the
domain of f and the subset of B consisting of points y such that y = f(x) for some x ∈ A
is called the range of f. We write f: A → B to mean f is a function fom A to B.
8.2. Continuity
Definition 8.6. A metric space is a pair (S, d), where S is a non-empty set and d: S x S → R
satisfies:
(i)
d(x, y) ≥ 0
for all x, y ∈ S,
18
(ii)
d(x, y) = 0
if and only if
(iii)
d(x, y) = d(y, x),
(iv)
d(x, z) ≤ d(x, y) + d(y, z)
x = y,
for all x, y, z ∈ S.
Definition 8.7. A mapping f from a metric space (S, d) into a metric space (T, p) is
continuous at x ∈ S if for any ε > 0 there exists δ > 0 such that d(x, y) < δ implies
p(f(x), f(y)) < ε.
The idea of continuity can be also applied to a specific kind of relations, i.e., to the
correspondences.
In the following definitions let S denote a subset of Rk, T is a subset of Rm, xn a
sequence of points of S, and yn a sequence of points of T.
Definition 8.8. Φ is a correspondence from S into T if for every x ∈ S there exists a nonempty image set Φ(x) which is a subset of T.
Definition 8.9. A correspondence Φ is upper semi-continuous at the point x0 ∈ S if
lim
n→∝
xn = x0, yn ∈ Φ(xn), lim
n→∝
yn = y0 implies y0 ∈ Φ(x0). A correspondence Φ is
lower semi-continuous at the point x0 ∈ S if lim n → ∝ xn = x0, y0 ∈ Φ(x0) implies that there
exists a sequence yn such that yn ∈ Φ(xn) and lim n → ∝ yn = y0. A correspondence Φ is semicontinuous at the point x0 ∈ S if it is both upper and lower semi-continuous at x0.
Lemma 8.1. Φ is an upper semi-continuous correspondence over S iff
graph Φ := {(x, y); x ∈ S, y ∈ Φ(x)} is a closed set in S × T.
Theorem 8.1. (Minkowski). Let K be a convex subset of Rm and z a point of Rm. There is
a hyperplane H through z and bounding for K if and only if z is not interior to K.
8.3. Fixed points
Consider a set S and a function f from S to S, i.e., a transformation of S into
itself. Great interest is attached to the existence of an element x’ such that x’ = f(x’), i.e.,
which coincides with its image, or which does not move in the transformation. Such an
element is called a fixed point of a transformation f.
One can prove the fundamental theorem.
Theorem 8.2. (Brouwer). If S is a non-empty, compact, convex subset of Rm, and if f is a
continuous function from S to S, then f has a fixed point.
19
The generalization of this result to correspondences from a set to itself plays an
essential role in the theory of general economic equilibrium. Consider now a set S and a
correspondence Φ from S to S. A fixed point of the correspondence Φ is an element x’
such that x’∈ Φ(x’), i.e., belongs to its image-set.
Now, one can prove:
Theorem 8.3. (Kakutani). If S is a non-empty, compact, convex subset of Rm, and if Φ is a
upper semicontinuous correspondence from S to S such that for all x ∈ S the set Φ(x) is
convex (non-empty), then Φ has a fixed point.
8.4. Exercises
I. Find the algebraic sum A + B of the following sets A, B ⊂ R2:
1. A = {(x,y) ∈ R2; 1 ≤ x ≤ 5, y = 0},
B = {(x,y) ∈ R2; x = 0, 0 ≤ y ≤ 4}.
2. A = {(x,y) ∈ R2; -1 ≤ x ≤ 3, y = 1},
B = {(x,y) ∈ R2; 4 ≤ x ≤ 6, 2 ≤ y ≤ 4}.
3. A = {(x,y) ∈ R2; y = x},
B = {(x,y) ∈ R2; x ≤ 0, y = 0}.
4. A = {(x,y) ∈ R2; 1/3 x ≤ y ≤ 2x},
B = {(x,y) ∈ R2; 3x ≤ y ≤ 4x}.
II. Verify that the function d(n1,n2) = 1/n1 - 1/n2, n1, n2 ∈ N is a metric in the set of
positive
intigers N. Find the ball B(7, 1/7).
III. In a discrete metric space (N, d1) find the fallowing balls: B(3, 5), B(4,1/5), B(1,1).
IV. Decide whether the respective function is a metric in R:
a) d(x,y) = (x - y)2, b) d(x,y) = x2 - y2, d(x,y) = x2 + y2, x, y ∈ R.
V. Decide whether the following points are interior/boundary points of the respective balls:
1. P(0, 8/10); B((0,0), 1) ⊂ R2.
2. P1(0,0), P2(0,3); B((3,3), 3) ⊂ R2.
VI. Give an example of an interior/boundary point of the following sets:
1. A = {(x,y,z) ∈ R3; x2 + y2 + z2 < 12}.
20
2. B = {(x,y,z,u) ∈ R4; x2 + y2 + z2 + u2 ≤ 4}.
VII. Use the definition to determine where the following correspondences S ⊂ R2 are
discontinuous:
1. S = {(x,y) ∈ R2; x+ y≤ 1} - {(1/2, 1/2)} ∪ {(x,y) ∈ R2; -3 ≤ x ≤ -1, 0 ≤ y ≤ 2}.
2. S = {(x,y) ∈ R2; 1 < x2 + y2 ≤ 9, 0 ≤ x} ∪ {(x,y) ∈ R2; x2 + y2 ≤ 1, x < 0}.
3. S = {(x,y) ∈ R2; 2 ≤ x ≤ 5, -x + 3 ≤ y ≤ x - 1} - {(4,3)} ∪
∪ {(x,y) ∈ R2; 2 < x ≤ 8, y = 3}.
4. S = {(x,y) ∈ R2; -1 ≤ x ≤ 0, 0 ≤ y ≤ 2} ∪ {(x,y) ∈ R2; 0 ≤ x ≤ 1, 1 ≤ y ≤ 3} ∪
∪ {(x,y) ∈ R2; 1 < x ≤ 5, -1 ≤ y ≤ -x + 6}.
VIII. Interpret Rl as the commodity/price space.
IX. Give an example of a correspondence S ⊂ R2 which is not upper/lower semicontinuous
at a point of its domain. Draw the graph of this correspondence and justify the respective
discontinuities by definition.
X. Give an example of a production set where the following assumptions are satisfied:
(i) continuity, possibility of inaction, convexity, free disposal, non-increasing returns to scale;
(ii) irreversibility, non-decreasing returns to scale, possibility of free production;
(iii) impossibility of inaction, irreversibility, additivity;
(iv) compactness, constant returns to scale.
Justify each answer.
XI. Verify the assumptions on the following production sets Yb and Y. Justify each answer.
(i) Y1 = {(y11, y21) ∈ R2; -3 ≤ y11 < 0,
-4 ≤ y21 < 0}
Y2 = {(y11, y22) ∈ R2; -1 ≤ y12 ≤ 1, y22 ≤ 2}
(ii) Y1 = {(y11, y21) ∈ R2; 1/2 y11 ≤ y21 ≤ -3y11}
Y2 = {(y11, y21) ∈ R2; 2y12 ≤ y22 ≤ -1/2 y12}
XII. Find the supply correspondences ηb, η and the profit functions πb, π of the following
production systems:
(i) B = {1,2},
Y1 = {(y11, y21) ∈ R2; -3 ≤ y11 ≤ 0,
-2 ≤ y21 ≤ 0}
Y2 = {(y11, y21) ∈ R2; y12 ≤ 0, -1 ≤ y22 ≤ 0}
(ii) B = {1},
Y = {(y1, y2) ∈ R2; 3y1 + y2 ≤ 0}
(iii) B = {1}
Y = {(y1, y2) ∈ R2; y2 ≤ -y1 for y1 < 0 and y2 ≤ -2y1 for y1 ≥ 0}
(iv) B = {1},
21
Y = {(y1,y2) ∈ R2; y1+ y2≤ 1}
XIII. Determine the preference relations
u: R2+ → R:
(i) u(x,y) = 2x + y
(ii) u(x,y) = 5xy
(iii) u(x,y) = min (x,y)
(iv) u(x,y) = x2y
Draw the respective indifference curves.
a
defined by the following utility functions
XIV. Find the budget set and the demand set of the consumer a with the consumption set
X(a) = R2+ and the following characteristics (the utility function ua, the initial endowment
e(a), and the price system p):
(i) u(x,y) = 3x + y, e(a) = (2,3), p = (1,1),
(ii) u(x,y) = xy, e(a) = (0,1), p = (1,2),
(iii) u(x,y) = x + 2y, e(a) = (2,2), p = (1,2),
(iv) u(x,y) = min (x,y), e(a) = (3,4), p = (4,2).
Draw the respective figures.
XV. Consider the Debreu private ownership economy E with the following characteristics:
l = 2, A = {1,...,10}, B = {1,2}, p = (1,1), xa* = (a,a+1) for a = 1,...,10,
Y1 = {(y11, y21) ∈ R2; -3 ≤ y11 ≤ 0, -4 ≤ y21 ≤ 0}, ω = (40,50).
(i) Determine the optimal production plans y1*, y2* of the producers such that the sequence
(x1*,...,x10*, y1*, y2*, p*) will be the equilibrium, where p* = p.
(ii) Give an example of the production set Y2 such that the equilibrium does not exist.
XVI. Consider the Debreu private ownership economy E with the following characteristics:
l = 2, A = {1}, B = {1,2}, p = (2,3),
Y1 = {(y11, y21) ∈ R2; -1 ≤ y11 ≤ 0, -2 ≤ y21 ≤ 0},
Y2 = {(y12, y22) ∈ R2; y22 ≤ -y12, y22 ≤ y12 + 3},
X = R2+, u(x1,x2) = x1 + 2x2, w = 10, θ11 = 1, θ12 = 0.
Find the total resources ω ∈ R2 such that the optimal actions of the agents will determine the
equilibrium.
XVII. Find an equilibrium in the Debreu private ownership economy E with the following
characteristics (I, J denote the sets of consumers and producers, respectively; p* = p):
1.
l = 2, I = {1,2}, J = {1}, p = (2,0), X1 = X2 = R2+, u1(x1,x2) = min (x1, 2x2),
Y = (-R2+) + {(1,0)}, ω = ω1 = (3,1), θ11 = 1.
2.
l = 2, I = {1,2}, J = {1}, p = (1,3), X1 = R2+, X2 = R2+ + {(-1,-1)}, u1(x1,x2) = x1 + 3x2,
u2(x1,x2) = 2x1 + x2, Y = (-R2+), ω1 = (2,1), ω2 = (1,4), θ1 = θ2 = 1/2.
3.
l = 2, I = {1}, J = {1,2}, p = (3,0), X1 = R2+ + {(-2,0)}, u1(x1,x2) = min (2x1, x2),
Y1 = (-R2+) + {(0,2)}, Y2 := {(y12,y22) ∈ R2; y12 ≤ 0, 2y12 ≤ y22 ≤ ½ y12}.
ω = ω1 = (2,2), θ11 = 1.
22
4.
l = 2, I = {1,2}, J = {1}, p = (0,3), X1 = R2+, X2 = R2+ + {(-1,1)}, u1(x1,x2) = x2 – x1,
u2(x1,x2) = 5, Y = (-R2+), ω1 = (2,3), ω2 = (1,1), θ2 = 1.
5.
l = 2, I = {1}, J = {1,2}, p = (1,2), X1 = [-2, +∞) × [0, 5] u1(x1,x2) = x2 – 2x1,
1
1
2
1
1
2
2
2
2
2
2
Y1 := {(y1 ,y2 ) ∈ R ; y2 ≤ -(1/2) y1 }, Y2 := {(y1 ,y2 ) ∈ R ; y1 ≤ 0, y2 = 3y1 },
ω1 = (2,4), θ1 = 1.
6.
l = 2, I = {1,2}, J = {1}, p = (5,0), X1 = [0, 10] × [0, 10], X2 = [-1, 7] × [0, +∞),
u1(x1,x2) = max (x1,x2), u2(x1,x2) = x1 – 2x2, Y = -X1, ω1 = (3,3), ω2 = (4,3), θ2 = 1.
{
}
7. l = 2 , I = {1, 2} , J = {1, 2} , Y1 = ( y1 , y 2 ) ∈ ℜ 2 : y 2 ≤ −2 y1 + 4, y 2 ≤ 4, y1 ≤ 3 ,
Y2 = [−3,0] 2 , p = (2,1) , X 1 = ℜ 2+ , X 2 = [0,+∞) × [0,4] , u1 ( x1 , x 2 ) = min{2 x1 , x 2 + 1},
1
2
u 2 ( x1 , x 2 ) = x1 + 2 x 2 , ω1 = (2,3) , ω 2 = (1,7) , Θ11 = , Θ12 = .
4
3
1


1 
l = 2 , I = {1, 2, 3}, J = {1}, p =  ,0  , Y1 = ( y1 , y 2 ) ∈ ℜ 2 : y 2 ≤ − y1 , y 2 ≥ −2 ,
2


2 
2
i∗
x = (i,2i ) for i = 1, 2, X 3 = [0,10] , u 3 ( x1 , x 2 ) = max{x1 − 1, x 2 + 1} , e(1) = (1, 7) ,
1
e(2) = (1,5) , e(3) = (6,6) , Θ1 = Θ 2 = .
4
8.
{
}
l = 2, I = {1, 2} , J = {1} , p = (1,0) , X 1 = ( x11 , x 12 ) ∈ ℜ 2 : 0 ≤ x11 ≤ 2, 0 ≤ x 12 ≤ 2 ,
9.
{
}
X 2 = ( x , x ) ∈ ℜ : 0 ≤ x ≤ 3, 0 ≤ x ≤ 1 , ω1 = (1,4) , ω 2 = (2,2) , u1 ( x , x ) = x11 x12 ,
1
u 2 ( x12 , x 22 ) = x12 ( x 22 + 1) , Y1 = ( y1 , y 2 ) ∈ ℜ 2 : −2 ≤ y1 ≤ 0, − 3 ≤ y 2 ≤ 0 , Θ1 = Θ 2 = .
2
2
1
2
2
2
2
1
2
2
{
1
1
1
2
}
1


l = 2 , I = {1, 2, 3} , J = {1} , Y1 = ( y1 , y 2 ) ∈ ℜ 2 : y1 − 3 ≤ y 2 ≤ − y1  , X 1 = ℜ 2+ ,
2


1
1
X 2 = [0,5] × [0,4] , X 3 = [0,+∞) × [0,5] , p = (3,0) , Θ1 = , Θ 2 = , ω1 = (3,5) , ω 2 = (2,7) ,
3
6
ω 3 = (4,1) , u1 ( x1 , x 2 ) = min{x1 − 2, x 2 } , u 2 ( x1 , x 2 ) = x1 x 2 and u 3 ( x1 , x 2 ) = x 2 − 3 .
10.
23
References
Barten A.P. and Boehm V. (1982), Consumer Theory, in: Handbook of mathematical
economics, K.J.Arrow and M.D. Intriligator, eds., Vol. II, chapter 9, p.381 - 429.
Elsevier.
Cournot A.A. (1838), Recherches sur les principes mathematiquesde la theorie des richesses,
Paris, Hachette; reprint: 1980, Ouvres de Cournot, vol. 8, Paris, J. Verin; English
transl.: 1927, New York, Macmillan.
Debreu G. (1959), Theory of value. Wiley.
Debreu G. (1982), Existence of Competitive Equilibrium, in: Handbook of mathematical
economics, K.J.Arrow and M.D. Intriligator, eds., Vol. II, chapter 15, p.697 - 743.
Elsevier.
Handbook of mathematical economics (1982), K.J.Arrow and M.D. Intriligator, eds., Vol. I III, Elsevier.
Malawski A. (1999), Metoda aksjomatyczna w ekonomii, Wrocław, Ossolineum.
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