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Mathematics for Economists 326

January 6, 2016

Mathematics for Economists 326 January 6, 2016 1 / 177
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Preliminaries

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Sets

A set is any collection of elements. Sets of objects will usually be denotedby capital letters, A, S, T for example, while their members by lower case,a, s, tfor example (English or Greek). A set S is a subsetof another set Tif every element ofS is also an element ofT. We write S T. IfS T,then x S x T.

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Sets

The set S is a proper subsetofT ifS T and S=T; sometimes onewrites S T orS T in this case. Two sets are equal sets if they eachcontain exactly the same elements. We write S=T whenever

x S x T and x T x S.

The upside down A,, means for all The backward E,

means there exists.

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Sets

A set S is emptyor is an empty set if it contains no elements at all. It is asubset of every set. For example, ifA={x| x2 = 0, x>1}, then A is

empty. We denote the empty set by the symbol. The complementof aset S in a universal set Uis the set of all elements in Uthat are not in Sand is denoted Sc. For any two sets S and T in a universal set U, wedefine the set differencedenoted S\T, as all elements in the set Sthat arenot elements ofT. Thus, we can think Sc =U\S.

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Sets

For two sets S and T, we define the union ofS and Tas the set

S T {x|x S orx T}.We define the intersection ofS and Tas the set S T {x| x S andx T}.

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Sets

The following are some important identities involving the operationsdefined above.

A

B=B

A, A

B=B

A (Commutative laws)

A (B C) = (A B) (A C), A (B C) = (A B) (A C)(Distribute laws)

(A B) C =A (B C), (A B) C=A (B C) (Associativelaws)

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Sets

The collection of all subsets of a set A is also a set, called the power setofA and denoted by

P(A). Thus, B

P(A)

B

A.

Example

Let A={a, b, c}. Then,P(A) ={, {a}, {b}, {c}, {a, b},{a, c}, {b, c}, {a, b, c}}.

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Sets

The order of the elements in a set specification does not matter. Inparticular,{a, b} ={b, a}. However, on many occasions, one is interestedin distinguishing between the first and the second elements of a pair. Onesuch example is the coordinates of a point in the x y-plane. Thesecoordinates are given as an ordered pair(a, b) of real numbers. Theimportant property of ordered pairs is that (a, b) = (c, d) if and only ifa=c and b=d.

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Sets

The product of two sets S and T is the set of ordered pairs in the form(s, t), where the first element in the pair is a member ofSand the secondis a member ofT. The product ofS and T is denoted

S T {(s, t)|sS, tT}.

The set of real numbers is denoted by the special symbol R and is definedas

R {x|

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Sets

Any n-tuple, or vector, is just an n dimensional ordered tuple (x1, . . . , xn)

and can be thought of as a point in n dimensional Euclidean space. Thisspace is defined as the set product

Rn R R

n times {(x1, . . . , xn)|xi R, i= 1, . . . , n}.

Often, we want to restrict our attention to a subset ofRn, called thenonnegative orthant and denoted Rn+, where

Rn+

{(x1, . . . , xn)

|xi

0, i= 1, . . . , n

} Rn.

Furthermore, we sometimes talk about the strictly positive orthant ofRn

Rn++ {(x1, . . . , xn)|xi >0, i= 1, . . . , n} Rn+.

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Functions

A function is a relation that associates each element of one set with asingle, unique element of another set. We say that the function f is a

mapping, map, ortransformation from one set Dto another set Y andwrite f :D Y.

We call the set D the domainand the set Y the codomainof the mapping.

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Functions

Ifyis the point in the range mapped into by the point x in the domain,we write y=f(x). In set-theoretic terms, f is a relation from D to Ywith the property that for each x

D, there is exactlyone y

Y such

that xfy (x is related to y via f).The rangeoff is the subset of the codomain that contains allf(x); that is,

range(f) ={y Y|x D s.t. f(x) =y}

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Functions

The imageof a subset S D under f is the set of points in the range intowhich some point in S is mapped, i.e.,

f(S) {y Y|y=f(x) for some x S}

The inverse image(or preimage) of a set of points AY defined asf1(A) {x D|f(x)A}.

The graph of the function f is the set of ordered pairs

G {(x, y)|x D, y=f(x)}.

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Functions

Iff(x) =y, one also writes x y. The squaring function s : R R, forexample, can then be written as s :x x2. Thus, indicates the effectof the function on an element of the domain. Iff :D Y is a functionand S D, the restriction off to S is the function f|Sdefined byf|S(x) =f(x) for every x S.

There is nothing in the definition of a function that prohibits more thanone element in the domain from being mapped into the same element in

the range.

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Sequences in R

A sequenceis a function k x(k) whose domain is the set{1, 2, 3, . . .}ofall positive integers. I denote the set of natural numbers by N =

{1, 2, . . .

}.

The terms x(1), x(2), . . . , x(k), . . . of the sequence are usually denoted byusing subscripts: x1, x2, . . . , xk, . . . . We shall use the notation{xk}k=1, orsimply{xk}, to indicate an arbitrary sequence of real numbers.

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Sequences in R

A sequence{xk} of real numbers is said to be1. nondecreasing ifxk xk+1 fork= 1, 2, . . .2. strictly increasing ifxkxk+1 fork= 1, 2, . . .

A sequence that is nondecreasing or nonincreasing is called monotone.

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Sequences in R: Convergence

DefinitionThe sequence{xk} converge to x if for every >0, there exists a naturalnumber N such that|xk x|< for all k>N. The number x is calledthe limit of the sequence{xk}. A convergent sequence is one thatconverges to some number.

Definition (Alternative definition)

The sequence

{xk

} converges to x if for every >0,

|xk

x

|< for all

but finitely many k.

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Sequences in R

Note that the limit of a convergent sequence is unique. A sequence thatdoes not converge to any real number is said to diverge. In some cases we

use the notation limk xkeven if the sequence{xk} is divergent. Forexample, we say that xk as k . A sequence{xk} is bounded ifthere exists a number M such that|xk| M for all k= 1, 2, . . . .

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Sequences in R

It is easy to see that every convergent sequence is bounded. The converseis true only for monotone sequences.

TheoremEvery bounded monotone sequence is convergent.

TheoremSuppose that the sequences{xk} and{yk} converge to x and y,respectively. Then,

1. limk(xk yk) =x y2. limk(xk yk) =x y3. limk(xk/yk) =x/y, assuming that yk= 0 for all k and y= 0.

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Exercise 1

limn

n+ 1 n=?Hint: (a b)(a+b) =a2 b2.

Exercise 2

limn

1 +

1

n

n e

Hint: Take logs and use the fact log(1) = 1

Exercise 3

xk= 2, xk+1=

xk+ 2

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Vectors

An n-vector a is an ordered n-tuple of numbers.

a= (a1, a2, . . . , an)

The operations of addition, subtraction and multiplication by scalars of

vectors are defined in the obvious way.

The dot product(or inner product) of the n-vectorsa= (a1, a2, . . . , an)and b= (b1, b2, . . . , bn) is defined as

a b=a1b1+a2b2+ +anbn =n

i=1

aibi

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Vectors

Proposition

If a, b, and c are n-vectors and is a scalar, then

1. a

b

=b

a

,2. a (b+c) =a b+a c,3. (a) b=a (b) =(a b).4. a a= 0 = a= 0

5. (a+b) (a+b) =a a+ 2(a b) +b b.

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Vectors

The Euclidean norm or length of the vector a= (a1, a2, . . . , an) is

a = a a=

a21+a22+ +a2n

Note thata =||a for all scalars and vectors.Lemma (Cauchy-Schwartz inequality)

|a

b

| a

b

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Vectors

The Cauchy-Schwartz inequality implies that, for any a, b Rn,

1 a bab 1.

Thus, the angle between nonzero vectors a and b Rn

is defined by

cos = a ba b , |0, ]

This definition reveals that cos = 0 if and only ifa

b= 0. Then

=/2. In symbols,aba b= 0


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Line and Hyperplane

Definition

The straight line in Rn

that passes through the point a and b is the set ofpoints{x| x=ta+ (1 t)bfor some t R}

Alternatively, we can eliminate tand write the equation for the line as

x1 a1b1 a1 =...=

xn anbn an

DefinitionThe hyperplane in Rn that passes through the point a= (a1, . . . , an) andis orthogonal to the nonzero vector p= (p1, . . . , pn), is the set of allpoints x= (x1, . . . , xn) such that

p (x a) = 0

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Exercise 4

What is the equation of the line through (1, 5, 3), (4, 6, 0)?

Exercise 5

What is the equation of the plane through (1, 1, 1), (-1, 1, -1), and (1, -1,-1)?

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Point Set Topology in Rn

Consider the n-dimensional Euclidean space Rn, whose elements, orpoints, are n-dimensional vectors x= (x1, . . . , xn). The Euclidean distanced(x, y) between any two points x= (x1, . . . , xn) and y= (y1, . . . , yn) in

Rn is the normx y of the vector difference between x and y. Thus,

d(x, y) =x y =

(x1 y1)2 +. . .+ (xn yn)2


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Ifx, y, and zare points in Rn, then

d(x, z)d(x, y) +d(y, z) (triangle inequality)Ifx0 is a point in R

n and ris a positive real number, then the set of all

points x Rn whose distance from x0 is less than r, is called the open ballaround x0 with radius r.

This open ball is denoted by Br(x0). Thus,

Br(x0) ={x Rn |d(x0, x)< r}


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DefinitionA set S Rn is open if, for all x0S, there exists some >0 such thatB(x0)S.

On the real line R, the simplest type of open set is an open interval. Let Sbe any subset ofRn. A point x0S is called an interior pointofS ifthere is some >0 such that B(x0)S. The set of all interior points ofS is called the interiorofS, and is denoted int(S).

A set S is said to be a neighborhoodofx0 ifx0 is an interior point ofS,that is, ifScontains some open ball B(x0) (i.e., B(x0)S) for some >0.


S
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Theorem

1. The entire spaceRn and the empty set

are both open.

2. The union of open sets is open.

3. The intersection of finitely many open sets is open.


P i S T l i Rn

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Each point in a set is either an interior point or a boundary point of the

set. The set of all boundary points of a set S is said to be the boundaryofSand is denoted S orbd(S). Given any set S Rn, there is acorresponding partition ofRn into three mutually disjoint sets (some ofwhich may be empty), namely;

1. the interior ofS, which consists of all points x

Rn such that N

S

for some neighborhood N ofx;

2. the exteriorofS, which consists of all points x Rn for which thereexists some neighborhood N ofx such that N Rn\S;

3. the boundary ofS, which consists of all points x

Rn with the

property that every neighborhood N ofx intersects both Sand itscomplement Rn\S.

The closureof a set S Rn is the union of the interior of the set and itsboundary,S= int S S. For a closed set, its closure is the set itself.


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Topology and Convergence

The basic idea to do so is to apply the arguments developed earlier to Rcoordinate-wise. A sequence{xk}k=1 in Rn is a function that for eachnatural number kyields a corresponding point xk in R

n.

DefinitionA sequence{xk} in Rn converges to a point x Rn if for each >0,there exists a natural number N such that xk B(x) for all k N, orequivalently, ifd(xk, x) 0 as k .


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DefinitionA set S

Rn is bounded if there exists a number M

R such that

x M for all x S. A set that is not bounded is called unbounded.Similarly, a sequence{xk} in Rn is bounded if the set{xk|k= 1, 2, . . .} isbounded.


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Theorem (Closed Sets)

A set S Rn is closed if and only if every convergent sequence of pointsin S has its limit in S.

The next concept ofcompact setsis used extensively in both mathematicsand economics.

DefinitionA set S in Rn is compact if it is closed and bounded.


C ti s F ti s
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Continuous Functions

Consider first a real-valued function z=f(x) =f(x1, . . . , xn) ofnvariables. Roughly speaking, f is continuous if small changes in theindependent variables cause only small changes in the function value.

TheoremA function f from S Rn intoRm is continuous at a point x0 in S if andonly if f(xk) f(x0) for every sequence{xk} of points in S thatconverges to x0.


Limits of functions
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Limits of functions

DefinitionL is a limit of function f :S R at x0S, L= limxx0f(x), if

xn x0 = f(xn) L

Corollary

A function f :S R is continuous at x0S if and only if

f(x0) = limxx0 f(x)


Calculus: Functions of a Single Variable

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Calculus: Functions of a Single Variable

The derivativesof function f() are defined as the limits

f(x) = limh0

f(x+h) f(x)h

,

f(X) = limh0

f

(x+h) f

(x)h

,

and so on.

In general, a function may or may not have a derivative. If a function

possesses a continuous derivatives f, f, . . . , fn, it is called n-timescontinuously differentiable, or a Cn function.


Calculus: Single Variable

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Calculus: Single Variable

Some rules of differentiation is provided below:

For constants, :d/dx= 0.

For sums: d/dx[f(x)

g(x)] =f(x)

g(x).

Power rule: d/dx(xn) =nxn1. Product rule: d/dx[f(x)g(x)] =f(x)g(x) +f(x)g(x)). Quotient rule: d/dx[f(x)/g(x)] = (g(x)f(x) f(x)g(x))/[g(x)]2.

Chain rule: d/dx[f(g(x))] =f(g(x))g(x).


Partial derivatives
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Partial derivatives

DefinitionLet y=f(x1, . . . , xn). The partial derivative off with respect to xi isdefined as

f(x)

xi lim

h0

f(x1, . . . , xi+h, . . . , xn) f(x1, . . . , xi, . . . , xn)h

y/xi or fi(x) orf

i(x) are also used to denote partial derivatives.


Gradients
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Gradients

Suppose that F(x) =F(x1, . . . , xn) is a function ofn variables defined onan open set A in Rn, and let x0 = (x01 , . . . , x

0n ) be a point in A. The

gradientofF at x0 is the vector

F(x0) =

F(x0)

x1, . . . ,

F(x0)

xn

of first-order partial derivatives.


Directional Derivative
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Directional Derivative

We define the derivative of f along the vector a by

fa(x) = limh0

f(x+ha) f(x)h

or, with components,

fa(x1, . . . , xn) = limh0

f(x1+ha1, . . . , xn+han) f(xi, . . . , xn)h

Usually, we normalize||a||= 1.


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Exercise 6

Find the derivative off(x1, x2) in the direction a= (1/2, 1/2)

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Level Surface
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S

DefinitionFor a function f : Rn R, and for a given x0 Rn, the level surfacethrough x0 is defined as

{x

Rn :f(x) =f(x0)

}.

DefinitionA curveon the surface through x0 is a mapping x : R Rn such that

f(x(t)) =f(x0), x(0) =x0


Tangent Hyperplane
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g yp p

DefinitionThe tangent hyperplaneis formed by the set of all tangent vectors x(t0)of the curves on the level surface through x0.

Applying the chain rule to f(x(t)) =f(x0), we see that the gradient

f(x0) is orthogonal to the tangent hyperplane.Corollary

The tangent hyperplane to the level surface at point x0 is given by

{x Rn

:f(x0) (x x0) = 0}


Gradient
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TheoremSuppose that f (x) =f(x1, . . . , xn) is C

1. Then, at points x where

f(x)= 0, the gradientf(x) = (f1(x), . . . , fn(x)) satisfies:1.f(x) is orthogonal to the level surface through x.2.f(x) points in the direction of maximal increase of f .3.f(x) measures how fast the function increases in the direction of

maximal increase.

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Econ 326 notes