A Ew Aproach to Teaching Mathematics in Engineering Education

7/31/2019 A Ew Aproach to Teaching Mathematics in Engineering Education

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A NEW APPROACH TO TEACHING MATHEMATICS IN

ENGINEERING EDUCATION

Uniqueness of the Maximum Point for Some Functions of Several Variables

Applied in the Calculation of Demand Function

Rodica-Mihaela Dne,professor,Department of Mathematics and Computer Science, Technical University ofCivil Engineering of Bucharest 124, Lacul Tei Blvd., 020396 Bucharest, Romania, e-mail: [email protected] Popescu,teaching assistant,Department of Mathematics and Computer Science, TechnicalUniversity of Civil Engineering of Bucharest 124, Lacul Tei Blvd., 020396 Bucharest, Romania, e-mail:[email protected]

Florica Voicu,associate professor,Department of Mathematics and Computer Science, Technical University ofCivil Engineering of Bucharest 124, Lacul Tei Blvd., 020396 Bucharest, Romania, e-mail: [email protected]

Abstract: The main purpose of this paper is to illustrate how the differential calculus courses canbe applied in science projects, for example in economic projects.We have written this paper to show that in some practical problems concerning extreme pointdetermination of certain functions of several variables it is possible to use only the necessaryconditions of extreme. So we have chosen a problem that previously showed existence anduniqueness of an extreme point for certain functions of several variables. We get these functions ineconomics, as functions that shape the demand for goods in some markets.

We have in view all markets of goods and services, including for example, the building materials orbuildings markets, and building equipment and service markets, or gas market, oil market and

electrical market.We analyze a Coob-Douglas preference and apply some mathematical facts as follows: sequences

in a -dimensionaln real space, functions of several variables and their continuity, graphs of such

functions, constrained extreme problems, -dimensionaln vectors, dot product function and its

(jointly) continuity, 1-norml .

Keywords: sequence, function of several variables, extreme point, demand function, preferencerelation, utility function.

Note that a small part of this paper is enclosed in[6] (an electronic issue).

Introduction

Preferences and utility functions, two basic concepts in economics, are not observable in themarket place. What we do observe are agents making transactions at market prices that is

demanding and supplying commodities at these prices.In this paper we will pay attention to the demand functions because these functions could be a

first alternative measure of economic behavior. An idea, used in literature (C. D. Aliprantis, D. J.Brown, O. Burkinshaw-1990), to model demand functions, is to derive such functions from

utility maximization subject to budget constraints. Consequently the demand functions satisfycertain restrictions which play a critical role in equilibrium analysis.

Many examples could be given, illustrating some known processes in a real market: As pricesgo to boundary, some goods become (relatively) cheap and, consequently, demand for some

commodities become (very) large and When prices drop to zero the demand collectively tendsto infinity. However it should be noted that when individual price of a commodity drops to zero,

the demand for that particular commodity does not necessarily tends to infinity(see [3]). Such


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examples can be formulated in particular markets, having two goods or having a finite number of

commodities.

We analyze Coob-Douglas preferences and apply some mathematical facts as follows: sequences

in a -dimensionaln real space, functions of several variables and their continuity, graphs of such

functions, constrained extreme problems, -dimensionaln vectors, dot product function and its

(jointly) continuity,1

-norml .

For a better understanding, we consider that this work should be self-contained. Therefore we

will begin the work with mathematics and economics preliminaries.

1. Mathematics Background

1.1. Binary Relations

In the sequel, X will be a (non-empty) set.

A binary relation (denoted by < ), on a set X is a non-empty subset of X X . For any,x y X we will denote by x y< the fact that ( ),x y belongs to < .

A binary relation < on a set X is said to bea). reflexive, ifx x< for all in X,

b). complete, if for any ,y in X, either y< ory x< holds,

c). transitive, ifx y< and y z< imply z< .

Such relations are used to describe the tastes of the economic agents.

Apreference on a set X is a reflexive, complete and transitive binary relation on .X

1.2. Topological concepts

The following notion describes mathematical structures that allow the formal definition ofconcepts as convergence and continuity.

A topological space is a (nonempty) set X , together with a collection of subsets of X ,

satisfying the following axioms:

a). the empty set and X are in ;

b). the union of any collection of sets in is also in ;

c). the intersection of any finite sets in is also in .

Sometimes we will denote ( ),X this topological space.

The collection is called a topology on X. The elements of are called open sets and their

complements in X are called closed sets. The elements ofX are calledpoints.

A neighborhoodof a pointx X is any set V X that contains an open set D containing(hence x D V ).

The neighborhood system ofx X consists of all neighborhoods ofx . Note that a topology onX can be determined by a set of axioms concerning all neighborhood systems.

A base (orbasis) B for a topological spaceX with the topology is a collection of open sets

in , such that every open set in can be written as a union of elements ofB .


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We say that the base B generates the topology .

Note that the notion of base for a topological space is very useful because many properties of atopology can be reduced to statements on a base generating the respective topology.

An interior pointof a subset A in a topological space X is a point x X such that there existsa neighborhood V ofx which is contained in A ( )x V A .

The interior of a subsetA in a topological space X is the set of all interior points of A and is

denoted usually by ( )int A .

We mention someproperties for the interior of a set:

a). ( )int A is an open subset ofA , even the largest open set contained in A ;

b). ( )int A is the union of all open sets contained in A ;

c). a set D X is open if and only if ( )intD D= ;

d). ( )int int intA A= ;

e). ifA B in X, then ( )int A is a subset of ( )int B ;

f). ifD is an open set in X, then D is a subset of the set A X if and only ifD is asubset of ( )int A .

A compact setin a topological space X is a set K X such that each of its open covers has a

finite sub-cover, meaning that for every arbitrary collection ( )A

D

of open subsets D

in X

such thatA

K D

there exists a finite subset J A with jj J

K D

.

A compact space is a topological space X such that X is a compact set.

Thefinite intersection property is the following property of a collection of subsets of a set X: if

( )i i IA =A is an arbitrary family of subsets of ,X then any finite sub-collection ( )j j JA ofA

(hence J is a finite subset ofI) has non-empty intersection (that is jj J

A

).

Note that the finite intersection property is useful in formulating an alternative definition of

compactness. Indeed, one can prove that: any topological vector space X is compactif andonly if every collection of closed sets satisfying the finite intersection property has nonempty

intersection itself.

A continuous function between two topological spaces ( )( )1,X X = and ( )( )2,Y Y = is afunction :f X Y such that the inverse image ( )1f D of every open set D Y ( )2D is

an open set in X, that is ( )1 1f D . (Remember that ( )1f D is the set ( ){ }X f x D | ).

1.3. Examples of Topological Concepts

We will use the terminology from [9], concerning topological concepts.

As a simple example of a topological space we quoteX = , the set of real numbers, endowedwith thestandardtopology, that is the base of this topology is the collection of all open intervals.


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More generally, the Euclidean spaces n ( )*n are topological spaces if we consider the(standard) topology, with the base equal to the collection of open balls.

Recall that on every n we can define a dot product ( 1 1 ... n nx y x y x y = + + , where

( )1,..., nx x x= and ( )1,...,n

ny y y= ), a norm ( x x= , for nx , this norm being called

theEuclidean norm) and a metric, or a distance function ( ( ),d x y x y= , for ,n

x y ); thenthe open (metric) ball of radius 0r> , centered at a point na , usually denoted by ( )rB a or

( ),B a r , is defined by ( ) ( ){ },nrB a x d x a r= such that for all x in S,

( ),d x a r < ).

The open interval ( )0,1 is not compact. Obviously, this is a consequence of the previous

theorem (Heine-Borel theorem), but we can prove directly, using the definition of a compact set;indeed the open cover

3

1 1,

nn n

does not have a finite sub-cover.

The space is not compact, because, for example the open cover ( )1, 1n

n n

+

has no finite

sub-cover.

All polynomial functions (that is the functions that can be defined by evaluating a polynomial,

for example ( ) 1 21 2 1 0...n n

n nf x a x a x a x a x a= + + + + + , for all arguments , where n is a non-

negative integer and0 1 2, , ,...,

na a a a are constant coefficients) are continuous. The rational

functions, exponential functions, logarithms, square root functions, trigonometric functions and

absolute value functions are continuous, too.


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1.4. Preference Relations and Utility Functions on Topological Spaces

The terminology and the notions used in this section appear, for example, in [3].

Apreference relation on a set X is a reflexive, complete and transitive binary relation.

Let < be a preference relation on X. For ,y X , x y< means that the bundle x is at least

as good as the bundle y or that x is no worse than y .

Also, x y , that is y< and y x< , means that x is preferred to y or that x is better thany .

Ifx y< and y x< , we say that is indifferent to y and denote this by ~x y .

Forx X , the better than set ofx is the set { }y X y x | and the worse than set ofx is the

set { }y X x y | . Analogous names are given to the sets { }y X y x | < and { }y X x y | < .

A preference relation < on a topological space X is said to be:

a). upper semi-continuous if for each x X , the set { }y X y x | < is closed;

b). lower semi-continuous if for each x X , the set { }y X x y | < is closed;

c). continuous, whenever< is both upper and lower semi-continuous, i.e. whenever for eachx X the sets { }y X y x | < and { }z X x z | < are both closed (or, equivalently, the

sets { }y X y x | and { }z X x z | are both open).

IfX is a set, a function :u X defines a preference relation on X, if the relations y<

and ( ) ( )u x u y are equivalent (where ,x y X ). Is said that u is a utility function representing

< .Obviously, the utility functions are not uniquely determined; indeed, if the function u represents

the preference relation < ; then the functions ( )u c c+ , ( )2 1nu n+ and ue represent < ,

too. But it is possible that no function exists that represents the preference relation < . The nextresult gives an answer to the question: When can a preference relation be represented by a

utility function?

Proposition 1.(see [3, Theorem. 1.1.4] and [8] for its proof) Every continuous preference on a

topological space with a countable base ( )1n n

B U

= of open sets can be represented by a

continuous utility function.

1.5. Vector Spaces

A vector space is a mathematical structure formed by a collection ofvectors that is objects thatmay be added together and multiplied (scaled) by numbers, called scalars in this context.

Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar

multiplication by complex numbers, rational numbers, or even more general fields instead. Theoperations of vector addition and scalar multiplication have to satisfy certain requirements,

called axioms, listed below.

We begin recalling the definition on a field, that is a set together two operations, usually called

addition and multiplication and denoted by + and , such that the following axioms hold:


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1. For all , in F, both + and are in F, or equivalently + and arebinary operations on F (F is closedunder addition and multiplication).

2. For all , , in F, the following equalities hold: ( ) ( ) + + = + + and

( ) ( ) = (associativity of addition and multiplication).

3. For all , in F the following equalities hold: + = + and = (commutativity of addition and multiplication).

4. There exists an element in F, called the additive identity elementand denoted by 0, suchthat for all in F, 0 + = . Likewise there exists an element, called the multiplicativeidentity element and denoted by 1, such that for all in F, 1 = . For technicalreasons, the additive identity and the multiplicative identity are required to be distinct, i.e.

0 1 . (additive and multiplicative identity).

5. For every in F, there exists in F such that ( ) 0 + = . Similarly, for any in

F other than 0, there exists an element 1 in F, such that 1 1 = (additive and

multiplicative inverses).

6. For all , , in ,F the following equalities hold: ( ) + = + (distributivity

of multiplication over addition).

Note that the elements ( ) + and 1 (for 0 ) are also denoted and

respectively. In other words, this means thatsubstraction and division operations exist. The most

commonly used fields are the field of real numbers, the field of complex numbers, and thefield of rational numbers. Any field may be used as the scalars for a vector space.

A vector space over a fieldF is a set V together with two binary operations, operations that

combine two entities to yield a third, called vector addition and scalar multiplication and that

satisfy some axioms. The elements of V are called vectors and are denoted in boldface (see

[15]). The sum of two vectors andv w is denoted +v w , the product of a scalar and a vectorv is denoted v orv . The axioms of a vector space are the following:

V1) ( ) ( )+ + = + +u v w u v w , for all , ,u v w in V (associativity of addition).

V2) + = +u v v u , for all u and v in V (commutativity of addition).V3) There exists an element 0 in V , called zero vector, such that + =u 0 u , for all u in V

(identity element of addition).

V4) For all u in V , there exists an element v in V , called the additive inverse of u , such that

+ =u v 0 (inverse elements of addition). Note that the additive inverse ofu is denoted u .V5) ( ) + = +u v u v , for all ,u v in V and in F (distributivity of scalar multiplications

with respect to vector addition).

V6) ( ) + = +u u u , for all u in V and , in F (distributivity of scalar multiplicationwith respect to field addition).

V7) ( ) ( ) =u u , for all , in F and u in V (compatibility of scalar multiplication withfield multiplication). Note that this axiom is not asserting the associativity of an operation,

since there are two operations in question, scalar multiplication (u ) and field multiplication

( ) .

V8) 1 =u u , where 1 denotes the multiplicative identity in F (identity element of scalarmultiplication).


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These axioms entail that substraction of two vectors and division by a (non-zero) scalar can be

performed via ( ) = + u v u v ,1

=

uu .

Remark. In contrast to the intuition stemming from vectors in the plane and higher-dimensional

cases, there is, in general vector spaces, no notion ofangles ordistances.

As examples of vector space we cite:

1. Coordinate space n ( n -dimensional realvectorspace).

2. Infinite coordinate space F (also called the space of infinite sequences, and also

denoted by ( )s ) ( ) ( ){ }*1,..., ,... , for all inn js x x x j= = x | .

3. Matrices space m n , with entries in and having m rows and n columns.

4. Polynomials vector spaces.

5. Functions spaces.

A linear functionalonn

is a function :n

f such that:

( ) ( ) ( )f f f+ = +x y x y , for all , nx y

( ) ( )f f =x x , for each nx and .

The set of all linear functionals on n , is denoted ( )'

n . Endowed with the pointwise operations

(that is ( )( ) ( ) ( )f g f g+ = +x x x and ( )( ) ( )f f =x x , for all nx , and

( )'

, nf g ), ( )'

n becomes a vector spaces, called the dualof n . We can see that, actually,

( )'

n n= .

The duality ( )( )'

,n n+ + was used in economics for the first time by G. Debreu ([5]) which

introduced the formal duality between commodities and prices. Later, C. D. Aliprantis and D. J.Brown ([2]) introduced the dual pair of (Riesz) topological spaces between commodity and price

spaces (see also [3]).

Today vector spaces are applied throughout mathematics, science and engineering. They are

appropriate linear algebraic notions to deal with systems of linear equations, offer a frameworkforFourier expansions, which are employed in image compression routines, or provide an

environment that can be used for solution techniques ofpartial differential equations.

Note that in this paper we work with ,nV = considered as a real vector space, that is as avector space over the field .F=

1.6. Convexity Concepts

IfE is a vector space, a set C E is said to be convex if for all x and y in C and all in

[ ]0,1 , the point ( )1 y + is in C. In other words, every point on the line segment

connecting and y is in C.


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A convex set A non-convex (i.e. concave) set

The convex subsets of (the set of real numbers) are the intervals of . Some examples of

convex subsets of the Euclidean two space 2 are regular polygons.

IfC E is a convex set, then for any 1,..., nx in C and any nonnegative numbers 1,..., n with

1 ... 1n + + = , the vector1

n

i i

i

x=

is in C. (A vector of this type is known as a convex

combination of 1,..., nx x ).

The intersection of any collection of convex sets is itself convex.

IfE is a vector space, a preference < defined on E is said to be:

a). Convex, whenever 1y x< and 2y x< in E and 0 1< < imply ( )1 21y y x + < ; for

all 0 1< < .

b).Strictly convex, whenever 1y x< and 2y x< in E and 1 2y y imply

( )1 21y y x + for all 0 1< < .

Obviously, a preference relation < defined on a convex set C is convex if and only if the set

{ }y X y x | < is convex, for all C .

A function : u C , defined on a non-empty convex subset C of a vector space, is said to be:

a). quasi-concave, whenever for each , y C with x y and each 0 1< < we have

( )( ) ( ) ( )( )1 min , ; + u x y u x u y

b).strictly, quasi-concave, whenever for each , x y C with x y and each 0 1< < we

have ( )( ) ( ) ( )( )1 min , ; + >u x y u x u y

c). concave, whenever for each , x y C with y and each 0 1< < , we have

( )( ) ( ) ( ) ( )1 1 ; + + u x y u x u y

d).strictly concave, whenever for each , x y C with x y and each 0 1< < , we have

( )( ) ( ) ( ) ( )1 1 ; + > + u x y u x u y

Note that a quasi-concave function : u C (with C a convex set) gives rise to a convexpreference, and a strictly quasi-concave function : u C gives rise to a strictly convexpreference (see [3, Theorem.1.1.8]).

Obviously, any concave function is quasi-concave, but the converse is false (a counterexample

being the function [ ): 0,u , ( ) 2u x x= ).

A function : u C , defined on a non-empty convex subset C of a vector space, is said to be:

a). convex, whenever the function u is concave, i.e. whenever for each ,x y in C andeach 0 1< < we have:


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( )( ) ( ) ( ) ( )1 1 ;u x y u x u y + +

b).strictly convex, whenever u is strictly concave.

A maximal elementfor a preference relation < on a non-empty subset A of a set X (if thereexists) is an element m A such that there is no element s A such that s m .

Note that an element m A is a maximal element for < on A if and only if m < for allA (this statement holds because < , as a preference relation, is complete).

It may happen that < need not have any maximal elements, but according to [3, Theorem 1.2.2],if X is a compact topological space and < is an upper semi-continuous preference relation,then the set M of all maximal elements for < is non-empty and compact. If moreover, < isconvex and A is a (non-empty) convex compactsubset of X then M is convex too, and if in

addition < isstrictly convex, then M has exactly one maximal element in X (see for example[3, Theorem. 1.2.3]).

1.7. Ordering Concepts

An order relation (an ordering) on a (non-empty) set E is a binary relation (on E) which is:

a). reflexive, that is x x for all in E;

b).antisymmetric, that is y and y x (for ,x y in E) imply that y= ;

c). transitive, that is x y and y z in E imply that z .We will denote also x y> if we have y and y .

An ordered vector space is a real vector space E endowed with an order relation satisfyingthe following properties connecting the algebraic and order structures:

1. if y ( ,x y in E) then x z y z+ + , for all z in E;

2. if y ( ,x y in E) and 0 in , then y .

Sometimes we will denote ( ),E instead ofE.

If ( ),E is an ordered vector space, thepositive cone ofE is the set { }0E x E x+ = | .

As the most known example of ordered vector space we mention nE= with the naturalordering

( ) ( )( )1 1where ,..., and ,...,n ny x x x y y y = =

if and only if j jx y

for all { }1,2,...,j n

.The positive cone of this ordered vector space is

( ) { }{ }1,..., 0, for all 1,2,...,n n jx x x j n+ = = | .

Note that y> in n if and only if j jx y for all { }1,2,...,j n and j jx y> for at least one j .

Apreference< on a non-empty subset M of an ordered vector space ( ),E is said to be

a).monotone if ,x y M and y> imply y< ;

b).strictly monotone if ,x y M and y> imply x y .


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Obviously, a strictly monotone preference is monotone, but the converse is false (a counter

example could be the preference relation on 2E += defined by the utility function

( ),u x y xy= ).

A level curve of a function : u C is any set ( ){ }C u x c =| , where c is any fixed real

number.A curve is said to be convex to the origin, whenever its graph has the shape shown in the

following figure, that is, ifA and B are any two points on the curve, then a ray

passing through the origin O and any point X of the line segment AB , will meet the curve atmost at one point D between O and X.

When does a function (defined on a convex set) have its

level curve convex to the origin?

The next result tells us that ordering and convexity

concepts are useful to obtain an answer to this question.

Proposition 2 (see [3, Theorem. 1.1.10]) Let Cbe a subset of the positive cone of an ordered

vector space and : u C a function which is strictly monotone and quasi-concave. Then thelevel curves ofu are convex to the origin.

An extremely desirable bundle (orvector) for a preference relation < , defined on a subset X of

a vector space E, is a vector v E

such that:

x v X+ for all X and all 0 > , and

x v x+ holds for all x X and all 0 > .

Obviously, if a preference relation has an extremely desirable bundle 0v > , then it has aninfinity of such vectors (because for all 0 > , v is an extremely desirable bundle too).

The following result is true, according, for example, to [3, Theorem. 1.2.1].

Theorem 3.If a preference relation < on n+ has no extremely desirable bundle then no

interior point of an arbitrary non-empty set nA+

can be a maximal element.

In the following theorem, topological, convexity and orderingconcepts are combined to give a

representation theorem for preferences defined on the positive cone n+ of a finite dimensional

vector space ( n ).

Theorem 4. (see [3, Theorem.1.1.12])If< is a continuous preference defined on n+ , then thefollowing holds:

1. If < is convex, monotone and has an extremely desirable bundle, then it can berepresented by a continuous, monotone and quasi-concave utility function;

2. If< is strictly convex and strictly monotone, then it can be represented by a continuous,strictly monotone and strictly quasi-concave utility function.


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1.8. Constrained Extreme for Functions of Two Variables and Lagrange Multipliers

Method

In mathematics, Lagrange multipliers method is used to find the maximum/minimum of a

function subject to constraints.

For example, consider the optimization problem:

( )maximize ,f x y

( )subject to ,g x y c=

We suppose that f is twice-differentiable and g is differentiable. We introduce a new variable

( ) , called aLagrange multiplier, and study theLagrange function defined by

( ) ( ) ( )( ), ; , ,F x y f x y g x y c = +

If ( ),y is a maximum for the original constrained problem, then there exists such that

( ), ,x y is a stationary (orcritical) point for the Lagrange function (stationary points are thosepoints where the first order partial derivatives ofF are zero1). But not all stationary points yield

a solution of the original problem. We need a sufficientcondition for optimality in constrained

problems. This condition is known as thesecond-derivative test. If ( ),M x y is a stationary point

for the Lagrange function F, we calculate 2 Md F| , having in view the constraint ( ),g x y c= .

This means that we use the equation 0dg= 0g g

dx dyx y

+ =

in the formula of

2

Md F| =2 2 2

2 2

2 22

M

F F Fdx dxdy dy

x x y y

+ +

|

If 2M

d F| 0> , then ( )f M is a minimum forM , and

if 2 Md F| 0< , then ( )f M is a maximum forM .

2. Economics Background

2.1. Preference Relations

It is quite natural to suppose that the economic agents act always to maximize their own welfareand therefore we can assume for each agent, the existence of a set of opportunities, from whichhe can consistently choose pairs of opportunities.

Then the tastes of each agent can be modeled by a preference relation. Indeed it is reasonable to

suppose, for example, that the taste of each agent is transitive; so, ifa is chosen over b and b is

chosen over c , then a will be chosen over c .

In a model of markets, namely the Arrow-Debreu model (see [2]), only a finite number of

commodities are exchanged, produced or consumed.

If there are n such commodities, it is natural to suppose that the commodity space is n .

1We emphasize that the notion ofstationary points can also be formulated if the functions f and g appearing in the definition ofF have n

variables, where 2.n >


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Indeed it is immediately that the commodity space is a vector space, because two commodity

bundles: can be added to produce a new commodity bundle and any scalar multiple of acommodity bundle is a commodity bundle.

Also the ordered vector space structure for this commodity space is necessary (see [10])

because, obviously the inputs for productions are negatively signed and the outputs of

productions are positively signed.

We also need a topological structure on the commodity space, because the linear operations ofvector addition and scalar multiplications must be continuous and the supply and demand

functions must depend continuously on prices; this last assumption must capture the economicintuition that a small change in prices results in a small change in demand and supply (see,

for example [3]).

We need also convexity concepts. To explain this we must discuss about the prices of

commodities. Given aprice vector ( )1,...,n

np p += p and a commodity vector ( )1,..., nx=x ,

the value of x at prices p is obviously1 1

...n n

p x p x+ + that is the dot product p x or the value

( )p x of a linear functionalp on the commodity space .n

Therefore, the price space can be view as the dual space of n , denoted by ( )'

n . But it is

known that ( )'

n n= . Now we ask:

What are the terms at which good j can be exchanged in the market for good i ? Obviously

these terms are defined by the ratio i

j

p

p, called the rate of substitution of the corresponding

prices. That is i

j

p

pis the amount of good j that can be exchanged for a unit amount of good i at

prices ( )1,..., np p=p .

Convexity is used (see [3, 1.1]) to express the behavioral assumption that the more an agent has

of commodity i , the less willing she is to exchange a unit of commodity j for an additional unit

of commodity i , i.e. convexity represents the notion of diminishing marginal rate of

substitution.

2.2. Demand Functions

Demand functions were introduced for the purpose of formulation of economic behavior without

using preferences and utility functions. The reason is that what we do observe in the marketplace are agents making transactions at market prices and that are demanding and supplying

commodities at these prices.

Note that, according for example to [3], we say that the vector nx isstrictly positive and wewill use the symbol 0x , if ( )1,...,

n

nx x= x is such that 0jx > for all { }1,...,j n . Also we

will denote x y if ( )1,...,n

nx x= x and ( )1,...,

n

ny y= y are such that j jy> for all

{ }1,...,j n .

Let ( )1,...,n

np p += p be aprice and let also ( )1,..., nw w=w be a vector in

n

+ .

The budget set forp corresponding to the vector w is the set


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( ) { }nB += w p x p x p w | Remember that the dot product function p x is 1 1 ... n np x p x+ + , if ( )1,..., np p=p and

( )1,..., nx x=x . This function (betweenn n

+ + and ) is (jointly) continuous and

consequently, the budget sets are closed.

The budget line of a budget set ( )Bw p is the set

( ){ }B = wx p p x p w| A question arises:

When does a price have compact budget sets?

Because we already established that each budget set is closed, and we know that in (a finite

dimensional vector space) n , compactness is equivalent to closeness and boundedness, we asknow

When does a price have bounded budget sets?

The following result shows us that either all budget sets for a price are bounded or else all areunbounded.

Proposition 5 (see [3, Theorem. 1.3.1])For a price n+p , the following statements hold.

1. All budget sets forp are bounded if and only if 0p .

2. All budget sets for p are unbounded if and only if p has at least one component equal to

zero.

Hence, the answer to our first question in this section is the following:

All budget sets for a price p are compact if and only if 0p .

As a consequence of this statement and of [3, Theorem. 1.2.3] (see also 1.6 in this paper), itfollows:

Proposition 6 (see [3, Theorem. 1.3.2])Let< be a continuous preference on n+ and 0p aprice. The following statements are equivalent:

i. If< is convex, then on every budget set for p, the preference < has at least one maximalelement;

ii. If < is strictly convex, then on every budget set for p, the preference < has exactly onemaximal element;

iii.If< has an extremely desirable bundle and is strictly convex, then on every budget set for p the preference < has exactly one maximal element lying on the budget line.

Here we have a geometrical interpretation of the

statement iii) in the previous result.

In the sequel, all preference relations will be

defined on n+ .

The interior of n+ , denoted by ( )int n+ , is

exactly the set of all 0x . A vector

n

+

y doesnot belong to ( )int n+ if it has at least one


14/27

component equal to zero. The set of all these vectors y is called the boundary of n+ and is

denoted by n+ .

Proposition 7 (see [3, Theorem. 1.3.3])For a price n+p and a preference relation < inn

+

the following are equivalent:

i. If the preference < is strictly monotone, then it does not have any maximal element in anybudget set forp ;ii. If the preference < is strictly monotone on ( )int n+ such that every thing in the interior is

preferred to any thing on the boundary and if an element n+w satisfies 0 >p w , then <

does not have any maximal element in ( )Bw p .

Now we will give the definition of a demand function. We consider a continuous strictly convex

preference relation < on n+ having an extremely desirable bundle. Also we fix a vector 0>w

in n+ and we call it the initial endowment. Then, according to Proposition 6, for any price

( )int n+p , the preference relation < has exactly one maximal element in the budget set

( )Bw p . This maximal element denoted by ( )wx p and even ( )x p is called the demand vectorof

the preference < at prices p .

The correspondence ( ) wp x p , between ( )int n+ and n+ is called the demand functionassociated to the preference < .

The most two importantproperties of the demand function are the following (see [3]):

1. Since (by Proposition 6 iii), ( )wx p lies on the budget line, for each ( )int n+p wealways have ( ) = wp x p p w .

2. The demand function is a homogeneous function of degree zero, i.e. for each 0 > andeach 0p we have ( ) ( )=w wx p x p . (This is a consequence of the budget identity

( ) ( )B B =w wp p .).

Because a demand function is defined for some preferences, it needs to give a name to these

preferences.

A neoclassical preference on n+ is a continuous preference relation < such that either

1. < is strictly monotone and strictly convex, or else;

2. < is strictly monotone and strictly convex on ( )int n+ , and everything in the interior ispreferred to anything on the boundary.

We remark that a neoclassical preference has always an extremely desirable vector x (indeed,

take any 0x ).

An importantproperty of the demand function corresponding to a neoclassical preference is itscontinuity. To prove this, we can use the following result.

Theorem 8 (The Closed Graph Theorem)Let :f X Y be a function between two topological

spaces, with Y compact and Hausdorff (that is for any two points 1 2,y y in Y , there exists two

neighborhoods 1V for 1y and 2V for 2y such that 1 2V V = ).


15/27

Then, f is continuous if and only if its graph ( )( ){ },fG x f x x X = | is a closed subset ofX Y .

We claimed that continuity is an important property of the demand function, because its

continuity expresses the fact that small changes in the price vector result in small changes in thedemand vector.

It is known that as prices go to the boundary, some goods become (relatively) cheap and

consequently demand for some commodities must become very large-see the following result.

Proposition 9 ([3, Theorem. 1.3.9]) Let < be a neoclassical preference on n+ and

( ) ( ) ( )( )1 ,..., nx x = x the corresponding demand function. Assume that the sequence ( )m mp ,with 0

mp for each m , converges to ( )1,..., mp p=p

(( ) ( )( ) ( )1 1

,..., ,..., ,m m

m n np p p p m= = p p ). Then it follows that:

1. If 0j >p for some j , then the sequence ( )( )j m mx p -thethj components of the demand

sequence ( )( )m mx p -is a bounded sequence .

2. If n+ p and 0 >p w , then ( ) ( )11

lim limn

m i mm m

i+ +

=

= = x p x p .

For other subjects concerning the demand functions see also [4], [7], [11], [12], [13].

3. An Example of Exchange Economy with Two Commodities

3.1. Preference Relation and Utility Function

In this section we will deal exclusively with the study of the demand function in a market with

two commodities.

We will consider that the preference relations (the taste of each consumer) on 2+ are

represented by Cobb-Douglas functions

( ) ( )( )1 2, , with 0< 1 2z z .

Note that 1 2z z means that 1z is preferred to 2z .


16/27

In the sequel, the vector space 2+ represents the commodity space of our economy

(where 2 is the number of available commodities).

3.2. Properties of the Preference Relation and Utility Function

1.The preference relation is strictly-monotone, that is2if , and , then+ >1 2 1 2 1 2z z z z z z .

Indeed, let ( ) ( )1 1 2 2= , and = ,x y x y1 2z z .Then we have the following:

( )

( ) ( ) ( )1 2 1 2 1 2

1 1

1 2 1 2 1 1 2 2

and y 1

2

x x y

u u x y x y

> > >

> >

z z

z z z z

and, obviously ( ) ( )1 2 .

2. The preference relation is strictly convex, that is and , with t z s z t s , imply that

( )1 , for all 0 1 + < t s t s , with t s and for each 0 1< < .Let us prove that u isstrictly quasi-concave.

Suppose that ( ) ( )u ut s that is 1 11 2 1 2t t s s for ( ) 21 2, ,t t += t ( )

2

1 2, ,s s += s t s

Denote 1 11 2 1 2, , ,t a t b s c s d = = = =

Then we have

2ad bc ad bc

+

(The geometric mean does not exceed the arithmetic mean.)

But ( )2

ad bc ab cd ab = and hence 2ad bc ab+

1 1 1

1 2 2 1 1 22t s t s t t + (3)

Now, because s t and ( )f x x= is strictly concave on ( )0, + ( ( )' 1f x x = and

( ) ( )'' 21 0f x x = < , for 0 1< < and 0x > ), it follows that:

( )( ) ( )1 1 1 11 1 0t s t s + > + > and ( )( ) ( )

2

1 1 1

2 2 21 1 0t s t s

+ > + > , for all

0 1< < .

So:

( )( ) ( )( ) ( )( ) ( )( )1 1 1

1 1 2 2 1 1 2 21 1 1 1t s t s t s t s

+ + > + + (4)

Using (3) and (4) we obtain

( )( ) ( )( ) ( )( ) ( )( ) (1 1

1 1 2 2 1 1 21 1 1 1u t s t s t s t

+ = + + > + +t s

( ) )121 s + ( ) ( ) ( )22 1 1 1 1

1 2 1 2 1 2 2 11 1t t s s t s t s = + + + >

( )2 2 2 11 21 2 2 2 t t > + + + ( ) ( )( )11 2 min ,t t u u = = t s .


17/27

Hence:

( )( ) ( ) ( )( )1 min ,u u u + >t s t s , for t s and for each 0 1< < .

Remark. The preference being continuous,strictly monotone andstrictly convex on 2+ , it is

a neoclassical preference. (Remember that a continuous preference on 2+ is said to be a

neoclassical preference whenevereither is strictly monotone and strictly convex, or else is

strictly monotone and strictly convex on ( )2int + and everything in the interior is preferred toanything on the boundary).

3.The preference relation is an upper semi-continuous preference on 2+ , that is for each

2

+z , { }2S += z w w z | is closed.But this means that if ( ),n n ns t S= zw (

*n ) and ( ) ( ),n s t n = w w , then S zw .

Indeed, if n S zw that is ( ) ( ), ,n ns t x y , or equivalently1 1

n ns t x y , for all *n and if

n w w , that is ns s and nt t ( )n , we obtain that 1 1s t x y that is S zw .

Note that if is continuous, it is represented by a continuous utility function, according to the

Proposition 1 in 1.4.

4.The preference relation has an extremely desirable vector(orbundle)

Indeed ( )1,1=v is such a bundle that is

a) 2 ++ z v , for all2

+z and 0 > ;

b) +z v z , for all 2+z and 0 > .

If ( ),x y=z , then ( ),x y + = + +z v and hence ( ) ( )

1 1

y x y

+ + >, for all

, 0x y > and 0 > ; that is ( ) ( ), ,u x y u x y + + > , +z v z for all 2+z and 0 > .

Remark. Because the utility function 2:u + is strictly monotone ( being a strictly

monotone preference) and (strictly) quasi-concave then its level curves are convex to origin (see

1.7).

5. Because the preference is an upper semi-continuous strictly convex preference then on any

compact subset 2C + , has exactly one maximal element in C (according, for example, to

[3, Theorem. 1.2.3]-see also 1.6 in this paper).

We remember that Cz is a maximal element for on C, whenever there is no elementCw satisfying w z . Since , as a preference relation is complete, Cz is maximal if and

only if z w for each Cw .

According, for example, to [3, Theorem. 1.2.1] all maximal elements of 2+ for the preference

lie in the same indifference set (in economics a level curve is known as an in-difference

curve) and if 2X += and has a strictly desirable bundle, then no interior point of any

A X can be a maximal element forA .

Now we suppose that the preference has the initial endowment1

1,2

=

w . In the sequel we

will calculate the demand function wx for .


18/27

For this aim, it must maximize the utility function ( ) 1,u x y x y = (with 0 1< < ) in the

budget line ( )Bw p :

( )1 2 1 2 1 21 1

1 1 02 2

p x p y p p p x p y

+ = + + =

.

We have a constrained extreme problem, where the constraint is given by the function

( ) ( )1 21

, 12

x y p x p y = +

We use theLagrange multipliers method. The Lagrange function is

( ) ( ) ( ) ( )1 1 21

, ; , , 12

F x y u x y x y x y p x p y

= + = + +

The necessary conditions for an extreme point of ( ), ;F x y are:

0F

x

=

, 0

F

y

=

, 0

F

=

(1)

The system (1) determines the parameter

and the coordinates of the points which eventuallyare extreme points for the function u . The system (1) becomes:

( ) ( )

( )

1 1

1

2

1 2

0

: 1 0

11 0

2

x y p

x y p

p x p y

+ =

+ = + =

S

For 0y , we can write the system ( )S as follows

( )

1

1

1

2

1 2

(2)

(3)1

11 0 (4)

2

px

y

px

y

p x p y

=

=

+ =

From (2) and (3), we obtain:

1

1 1

2 2

1 1p pxy x

y p p

= =

(5)

From (4) and (5), it follows that

( ) 11 22

1 11 0

2

pp x p x

p

+ =

1 1 21 1 1 2

21 1

2 2

p p pp p x p p x

+ + = + =

1 2

1

2

2

p px

p

+= (6)

Now (5) and (6) imply:

( ) 1 22

21

2

p py

p

+

= (7)


19/27

The stationary point for u with the constraint is ( ),y given by (6) and (7). It is not

necessary to apply thesufficient condition for this constrained extreme because from the theory(see 1.6) we know of the existence of a unique maximal element for our constrained problem,

and, consequently this element is exactly ( ),y given by (6) and (7).

Hence, the demand function corresponding to the preference is

( ) ( )1 2 1 21 2

2 2, 1

2 2

p p p p

p p

+ +=

wx p (8)

or

( ) ( )( )

( )1

2 , 22 2

t t tt

= + +

wx , if

2

1

pt

p= .

We remark that the first two properties of the demand function are valid. Note that these

properties were mentioned in section 2.2:

1) Since ( )wx p lies on the budget line, for each ( )2int +p , we always have

( ) = wp x p p w

2) The demand function is a homogeneous function of degree zero, i.e. for each 0 > and0p , we have ( ) ( )=w wx p x p .

Moreover, we remark that our wx is obviously continuous (in generally this follows from

Theorem 8, becausew

x corresponds to a neoclassical preference).

So, small changes in the price vector result in small changes in the demand vector. The

geometrical meaning of this statement is depicted in this figure.

As the price vectorp changes to q , the demand vector ( )x p changes to ( )x q .

If ( ) ( ) ( )1 2 1 21 2

2 2, 1

2 2

p p p p

p p

+ += =

wx p x p is the demand vector, then the real number

( )1 2 1 21 2

2 21

2 2

p p p p

p p

+ + + ,

denoted by ( )1

x p (the1l-norm of ( )x p , because the components of ( )x p are positive!)

represents the total number of units of goods demanded by the (individual) consumer.


20/27

Remark. For an arbitrary initial endowment ( )' '', 0w w= >w , we find the demand function of , if we maximize the utility function ( ) 1,u x y x y = (with 0 1< < ) in the budget constraint

( )Bw p :

' ''

1 2 1 2p x p y p w p w+ = +

or equivalently

( ) ( )' ''1 2 0p x w p y w + = .We denote the constraint by

( ) ( ) ( )' ''1 2,x y p x w p y w = + .TheLagrange function is

( ) ( ) ( ), ; , ,F x y u x y x y + .

So, the necessary conditions for the extreme point are the following:

0F

x

=

, 0F

y

=

, 0F

=

We obtain like in the previous particular problem:

1

2

1 py x

p

= ,

and hence

( ) ( )' ''1 2 0p x w p y w + = implies ( )' ''11 22

10

pp x w p x w

p

+ =

( )1 2 ' ''1 21p p

x p w p w

+ = + or

' ''

1 2

1

p w p wxp

+= or, equivalently1

xp= p w .

Then, it follows that

2

1y

p

= p w

Hence, the demand function for the preference is

( )1 2

1,

p p

=

wx p p w .

A new question arises:

When did the demands for the two commodities are equal?

Obviously, if 0 >p w , then:

( ) ( )=x p y p if and only if1 2

1

p p

= 1

2

p

p = 2

1 2

or 1p

p p

=

+ , that is

( )1 ,1 =

p

p .


21/27

According to Proposition 9 (from 2.2), because is a neoclassical preference on 2+ , if

( ) *n np is a sequence of strictly positive vectors such that( ) ( )( ) ( )1 2 1 2, ,n nn p p p p= =p p

Then, the following are true:

1. If1

0p > or2

0p > , then the two components ( )nx p and ( )ny p of the demand sequence

( )( )n nwx p are bounded sequences. Indeed, if for example ( )1

1, 1,0nn

=

p ( n ),

then ( ) ( )' '' 'n nw w wn

= + wx p x p is a convergent sequence and, therefore it is a

bounded sequence.

2. If ( ) 21 2,p p += p (that is 1 0p = or 2 0p = ), and' 0 >p w , then

( ) ( ) ( )( )1

lim limn n nn n

= + = w

x p x p y p

In other words, if the prices drop to zero, then the demand collectivity tends to infinity . Indeed,

if, for example, ( ) 21

1, 1,0nn

+

=

p , then

( ) ( )( ) ( )( )' ''1

lim lim 1 1n nn n

w w nn

+ = + + =

x p y p .

For this example, if the individual price of the second commodity drops to zero, the demand

( )ny p for this commodity tends to infinity, because

( ) ( )' ''11 1n w w nn

= +

y p

In general it is possible that in spite of the fact that the collectively demand tends to infinity andthe price of a particular commodity converges to zero, the demand for this single commodity can

be bounded.

4. An Example of Exchange Economy with a Finite Number of Commodities

Now we generalize the example in the previous section, switching to a market with more goods.

We will simplify the method used (the case of the market with two goods) for solving the systemleading to a set of stationary points of the utility function that describes a preference relation. Wewill use the observation (see section 1.7) that a preference relation can be described by several

utility functions and we replace the Cobb-Douglas utility function by its left composition with a

logarithmic function. Thus we have linearized the problem to be solved. Hence we will consider

that the preference relations (the taste of each consumer) on n+ are represented by Cobb-

Douglas functions

( ) 1 21 2 1 2, ,..., ... nn nu x x x x x x =

with 1 1

n

ii = = and 0 1

i< < , for each { }1,..., .i n


22/27

Hence the preference relation associated with u is defined by

( ) ( )( )1 2 1 2, ,..., and , ,...,n nx x x y y y= = x y x y 1 2 1 21 2 1 2... ...n nn nx x x y y y

It is immediately that if u is a utility function for the preference and f is a monotone

function (increasing that is ( ) ( )f f x y x y ordecreasing that is ( ) ( )f f x y x y ),

then 1u f u= is also a utility function describing the preference .

For example we can use the logarithmic function lnf = . Thus, the utility function for the

preference is ( ) ( )1 21 1 2 1 2 1 1 2 2, , ..., ln ... ln ln ... ln .nn n n nu x x x x x x x x x = = + + +

Remember that a set nC is said to be convex if for all x and y in C and all in [ ]0,1 , the

point ( )1 + x y is in C. In other words, every point on the line segment connecting x and

y is in C (see 1.6, in this paper).

Moreover note that a set nC is compact(see 1.2) if it issequentially compact, that is every

sequence ( )m mx in C has a convergent subsequence ( ) ( )1 2 ... .mk mm k k k x Remember that a

sequence ( )m mx inn is convergent to the limit nx , where ( ),1 ,2 ,, ,...,m m m m nx x x=x , for each

( )1 2 ,1 1 ,, and , ,..., , if ,..., when .n m m n nm x x x x x x x m = x

Remember also that a set nF is said to be closed(see 1.2) if for any convergent sequence in

F, its limit, calculated in n , is in F, too.

1. Firstly we will prove that if the preference is described by the utility function 1u defined

on a convex compact set nC + , it has the following properties.

1) is upper semi-continuous on C, that is, for any Cx , the set { }C C= x y x y isclosed.

2) is strictly convex on C, that is, if y x and x z , and in Cy z , then

( )1 + y z x for all 0 1.< <

(remember that ( )1 + y z x means that ( )1 + y z x and ( )1 + x y z )

We mention that is strictly convex if and only if the utility function 1u is strictly quasi-

concave that is for any x and y in C with x y and any 0 1,< < we have

( )( ) ( ) ( )( )1 1 11 min ,u u u + x y x y

Indeed, we have the following:

1) Let nx C + and ( ) ,n

m mmy C y y + x , when .m

Suppose that ( )1 2, ,..., nx x=x , ( ),1 ,2 ,, ,...,m m m m ny y y=y and ( )1 2, ,..., ny y y=y .

Hence ,1 1 ,,..., when .m m n ny y y y m Also, because m C xy for any m , we have

my x , or, equivalently


23/27

( ) ( ) ( ) ( )

( ) ( )1 2 1 21 1

,1 ,2 , 1 2

1 ,1 , 1 1

, ln ln ,

ln ... ln ... ,

ln ... ln ln ... ln , .

n n

m m

m m m n n

m n m n n n

u u m u u m

y y y x x x m

y y x x m

+ + + +

y x y x

But ,1 1 ,,..., whenm m n ny y y y m and hence, by using the algebraic operations with the

convergent sequences in n , and the fact that the logarithmic function is continuous, we obtain

( )1 ,1 , 1 1ln ... ln ln ... lnm n m n n ny y y y m + + + + (2)

By taking the limits ( )m in the two sides of the inequality (1), and by using (2) we obtain:

1 1 1 1ln ... ln ln ... lnn n n ny y x x + + + + that is

( ) ( )1 2 1 21 2 1 2ln ... ln ...n nn ny y y x x x or, equivalently ( ) ( )ln lnu uy x (3)

But the function lnf = is increasing and hence ( ) ( )u uy x ory x that is C xy .

So, we proved that set Cx is closed, for each .Cx

2) Let x and y in C with x y and 0 1.< <

We have to prove that

( )( ) ( ) ( )( )1 1 11 min ,u u u + x y x y (4)

(Because nC + is a convex set, and x and y in C, then for each 0 1< < it follows

that ( )1 C + x y ).But ( ) ( )( )1 lnu u=x x , for any .Cx

Then, because lnf = is an increasing function, it follows that (4) is equivalent with thefollowing inequality

( )( ) ( ) ( )( )1 min ,u u u + x y x y (5)

Putting ( )1 2, ,..., nx x x=x and ( )1 2, ,..., ny y y=y (5) becomes:

( )( ) ( )( ) ( )1 1 2 1 2

1 1 1 2 1 21 ... 1 min ... ; ...n n nn n n nx y x y x x x y y y

+ + (6)

We mention that (6) was proved in section 3 of the paper, for 2n = . For an arbitrary n we canuse the mathematical induction.

2.Secondly, we notice that according to the following theorem, mentioned (in a more general

version) in 1.6 in this paper, our preference has exactly one maximal element in C.

Theorem 10. Let nC be a convex compact set and let be an upper semi-continuous

preference relation on C. LetM be the set of all maximal elements for (Remember see-1.6

that an element c C is said to be maximal for , ifc , for allx C ). Then the following

statements hold:a) The setM is non-empty, convex and compact.


24/27

b) If, in addition is strictly convex, then M has exactly one element, that is the preference

relation has exactly one maximal element in .C

3.Now we will calculate the (unique) maximal pointfor . Remember that the utility function

for the preference relation is ( ) ( )1 21 1 2 1 2, ,..., ln ... nn nu x x x x x x = = . Suppose that the initialendowmentis ( )1,..., .nw w=w We mention that the convex compact set C is here the budget set

( )Bw p , where ( )1,..., np p=p is theprice-vector. It must maximize the utility function 1u in the

budget line of ( )Bw p , that is in the set

( ){ }1 2 1 1 1 1, ,..., ... ...n n n n nx x p x p x p w p w= + + = + +x

we have a constrained extreme problem, where the constraintis given by the function

( ) ( ) ( )1 2 1 1, ,..., ... .n n nx x x p x w p x w = + +

We use theLagrange multiplication method.

TheLagrange function is ( ) ( ) ( )1 1 2 1 1 2 1 2, ,..., ; , ,..., , ,..., .n n nF x x x u x x x x x x = +

The necessary conditions for an extreme point of ( )1 1 2, ,..., ;nF x x x (see footnote in section 1.8)

are the following:

1 1 1

1 2

0, 0,..., 0, 0n

F F F F

x x x

= = = =

(7)

(remember that ( )1 1 1 1,..., ln ... lnn n nu x x x x = + + )

We obtain the following system:

( ) ( )

1 21 2

1 2

1 1 1

0; 0; ...; 0 (8)

... 0 (9)

nn

n

n n n

p p px x x

p x w p x w

+ = + = + =

+ + =

Because from (8) it follows thatj

j

j

xp

= , for each { }1,2,...,j n , the equality (9) becomes:

11 1

1

... 0

+ + =

nn n

n

p w p wp p

( )11 1 11

... ... 0

+ + + + =

nn n n

n

p p p w p wp p

( )1 1 11

... ...

+ + = + +n n np w p w

But 1 ... 1 + + =n . Then if follows that1 1

1

...

=

+ + n np w p w

and hence


25/27

1 1...

+ +

= n nj jj

p w p w

p(10)

for each { }1,2,...,j n .

Therefore, the stationary points of 1u , with the constraint is ( )1,..., nx x given by (10).

It is not necessary to apply the sufficient conditions for this constrained extreme because from

theory (see Theorem 10 above) and the properties of function 1u we know the existence of a

unique maximal elementfor this problem, and, consequently, this element is exactly ( )1,..., nx

given by (10).

4. This maximal element is known as a demand vector, and the function

( ) 1 1 1 111

... ...,...,

+ + + +=

n n n nw n

n

p w p w p w p wx p

p p

where ( )1,...,= np pp and ( )1,...,= nw ww is the demand function corresponding to the

preference .

5.Historical note (Cobb-Douglas utility functions)

In economics this function was proposed by Knut Wicksell (1851-1926) and tested against

statistical evidence by Charles Cobb and Paul Douglas (1900-1928).

Charles Wiggans Cobb (1875-1949) was an American mathematician and economist. He

published many works on both subjects. However he is most famous for developing the Cobb-Douglas formula in economics. He worked on this project with the economist Paul Douglas

while lecturing at the Amherst College in Massachusetts.

Paul Howard Douglas (March 26, 1892-September 24,1976) was an American politician andUniversity of Chicago economist. (He served as a Democratic U.S. Senator from Illinois from

1949 to 1967)

The Cobb-Douglas utility function (with two variables) is

( ) ( )1 2 1 2, 1 = + =u x x x x

were 1x and 2 are the quantities consumed of good 1 and good 2.

In its generalized form, the Cobb-Douglas utility function is

( ) 1 21 2 ... = nnu x x x x

where ( )1,...,= nx x ,1

1=

=n

i

i

and 0 1<


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References

[1] Arrow, K. J.: An extension of the basic theorems of classical welfare economics, in J. Neymann Ed.,Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Univ. of CaliforniaPress, Berkeley and Los Angeles (1957), 507-532.

[2] Aliprantis, C. D.; Brown, D. J.: Equilibria in markets with a Riesz space of commodities, J. Math. Econom.11(1983), 189-207.

[3] Aliprantis, C. D.; Brown, D. J.; Burkinshaw, O.: Existence and Optimality of Competitive Equilibria, Berlin,Heidelberg, New York, London, Paris, Springer-Verlag, 1990.[4] Codenotti, B.; Melune, B.; Varadajan, K.:Market equilibrium via the excess demand function, Proceedings of

37th annual ACM Symposium (2005), 74-83.[5] Debreu, G.: Variation equilibrium and Pareto Optimum, Proc. Nat. Acad. Sci. USA, 40(1954), 588-592.[6] Dne, R.-M.; Popescu, M.-V.; Voicu, F.:How we can use the differential calculus to compute the demand in

an arbitrary market, A XVI-a Conferin, Confort, Eficien, Conservarea energiei i Protecia mediului (CECEPM - XVI), Technical University of Civil Engineering of Bucharest, Faculty of Building Services, 18-19March 2010, Proceeding of the Conference, electronic issue, ISSN 1842-6131.

[7] Fare, R.; Gross Kopf, S.; Hayes, K.; Margaritis, D.: Estimating Demand with Distance Functions:Parametrization in the Primal and Dual, Journal of Econometrics, 147(2)(2008), 266-274.

[8] Hildebrand, W.; Kirman, A. P.:Introduction in Equilibria Analysis, Advanced Textbooks in Economics #6,North Holland, Amsterdam, 1976.

[9] Kelly, J. L.: General Topology, D. Van Nostrand Co., New York & London, 1955.[10]Kreps, D. M.: Arbitrage and equilibrium in economics, with infinitely many commodities, J. Math. Econ.

8(1981), 15-35.[11]Singh, R.; Kumar, S.: Application of the Alternative Techniques to estimate Demand for Money Developing

Countries, The Journal of Developing Areas, 2009, mpra. ub. uni-muenchein. de/19295/.[12]Shafer, W.; Sonnenschein, H.: Market demand and excess demand functions, Handbook of Mathematical

Economics, 1993.[13]Soon, W.; Zhao, G.; Zhang, J.: Complementary demand functions and pricing models for multi-product

markets, European Journal of Applied Mathematics, 20(2009), 399-430, Cambridge Univ. Press.[14]Wikipedia: www.en.wikipedia.org

[15]Waerden, van der; Leendert, B.:Algebra (9 th ed), Berlin, New York, Springer-Verlag, 1993.


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