Important Mathematical Tools
Learning objectives:
.
.
.
.
Ability to apply important mathematical tools for economic
analysis
Ability to give interpretations of mathematical properties in
terms of economic concepts
• Topics reviewed
– Sets: definition, notation
– Binary relations, orderings
– Functions, sequences
– Sets: properties
– Brower’s fixed point theorem
– Separation theorems (Minkowski)
c Ronald Wendner
GE-Math-1
v2.1
1
Sets: Definition, Notation
• set
notation: upper case letter
use elements to define set: X ≡ {x | P (x)}
X ≡ {x ∈ students at KFUG | x study economics}
• elements of a set: x ∈ X, x 6∈ X
most sets we use are sets of vectors (w/ an interpretation)
elements: points or vectors
N ≡ {x | x is a natural number}, -1.3 6∈ N
R ≡ {x | x is a real number}, -1.3 ∈ R
R+ ≡ {x | x is a nonnegative real number}, -1.3 6∈ R+
R++ ≡ {x | x is a strictly positive real number}, 0 6∈ R++
∅ empty set (null set)
notice that ±∞ 6∈ R
• subsets Xi ⊂ X iff for all x ∈ Xi: x ∈ X
masters students in econ are a subset of students in econ
∅ ∈ X (!)
• set operations/relationships & logical connectives
–
–
–
–
–
subset relation; set equality
set union, intersection, complement
set subtraction
(∀x ∈ A)(x ∈ B) , (∃x ∈ A)(x ∈ B)
(Cartesian) set product
c Ronald Wendner
GE-Math-2
v2.1
• Cartesian product, cross product, direct product
– set ordered n-tuples
C ≡ A × B ≡ {(a, b) | a ∈ A , b ∈ B}
(4, 5) ∈ C, (5, 4) ∈ C, (4, 5) 6= (5, 4)
Example 1. L = 2, p = (p1, p2):
pl ∈ R+, p = (p1, p2) ∈ R+ × R+ = R2+, (1, 2) 6= (2, 1)
Example 2. A = B = [0, 1]. What is C = A × B?
Example 3. Suppose we have L commodities. Then every commodity bundle is an element (a vector) of:
RL+ ≡ R+ × R+ × R+ × .... × R+ (L-fold Cartesian product)
• dot product notation:
– inner (dot) product of two L-vectors
x = (x1, x2, ..., xL), y = (y1, y2, ..., yL):
 
y1
L
  X
 y2 
x · y ≡ (x1, x2, ..., xL)  ..  =
xl y l .
.
l=1
yL
Example: x = (x1, x2, ..., xL), p = (p1, p2, ..., pL);
cost of consumption bundle x is: p · x
c Ronald Wendner
GE-Math-3
v2.1
2
Binary Relations, Orderings
Consider two elements of a set: a, b ∈ X.
• binary relations (R): a R b
–
–
–
–
–
–
–
a is the brother of b
a is to the left of b
a>b
a≥b
a=b
a%b
ab
• quasi-ordering
– reflexivity: a R a for all a ∈ X
– transitivity: a R b, and b R c ⇒ a R c.
• complete ordering
for all a, b ∈ X, either a R b or b R a or both
how to compare Pizza with Rogan Josh?
• Rational % are a complete ordering (binary relation)
on consumption space
c Ronald Wendner
GE-Math-4
v2.1
• upper contour sets of % on X
X +(y) ≡ {x ∈ X | x % y}
• lower contour sets of % on X
X −(y) ≡ {x ∈ X | y % x}
• indifference set of % on X
X −(y) ∩ X +(y)
3
Functions, Sequences
• A function f : A → B is a rule that associates to every
element of set A exactly one element in set B. Set A is called
the domain of the function, and set B is called the codomain
of the function.
→ onto if for every y ∈ B, ∃ x ∈ A | y = f (x)
→ one to one if for any two elements in the domain, x, x0 ∈ A
with x 6= x0, it holds: f (x) 6= f (x0)
Example 1. Consider f (x) : R+ → R+, with f (x) = x2. Then
f (x) is onto and one to one.
Example 2. f (x) : R → R+, with f (x) = x2. Then f (x) is
onto but not one to one
c Ronald Wendner
GE-Math-5
v2.1
• Sequence f : N → R
– elements: xn, n = 1, 2, ....;
sequence: {xn}∞
n=1
– sequences in R: 1/n, or n/(1 + n), or 5
– sequences in R2: (1/n, n/(1 + n)), or (1/n, 5)
– “sequence in set X”: xn ∈ X for all n = 1, 2, ...
– convergence: limn→∞ xn = x
∗ for all > 0, ∃ n0 ∈ N : ∀n > n0, d(xn, x) < ∗ “convergence in set X”: limit x ∈ X
{xn}∞
n=1 = 1/n, A = [0, 1], B = (0, 1]
• subsequence
m(n) = strictly increasing function, m(n) : N → N
e.g., m(n) = 3 n
n ∞
{xm(n)}∞
n=1 is subsequence of {x }n=1
Theorem 1 (Bolzano-Weierstrass) .
Every bounded sequence in RL contains a convergent
subsequence.
c Ronald Wendner
GE-Math-6
v2.1
• Continuity of a function
Let f (x) : X → R, X ⊆ RL. f (x) is continuous at x if for
every sequence {xn} → x, {f (xn)} → f (x).
f (x) is continuous if function is continuous at all x ∈ X.
→ if f (.) is continuous: image of compact set is compact
Theorem 2 (Extreme value theorem) .
If f (.) is continuous and X compact then f (.) has a maximum/minimum.
→ compactness? ...
4
Sets in Euclidean space RL: Properties
• Euclidean distance d(x, x0)
0
L
0
– x, x ∈ X ⊆ R , d(x, x ) ≡
qP
L
l=1 (xl
− x0l )2
• -neighborhood (nbhd) about x:
N(x) = {x0 ∈ X | d(x, x0) < }
• open set
X ⊂ RL open if for every x ∈ X there exists an > 0 such
that N(x) ⊂ X.
c Ronald Wendner
GE-Math-7
v2.1
• closed set
X ⊂ RL closed if every converging sequence in X
converges in X
consider X = (0, 1], {xn}∞
n=1 = 1/n
example: closed interval: [a, b] ⊂ R ≡ {x ∈ R | a ≤ x ≤ b}
→ closed sets and maximizing bahavior
• bounded set
X bounded if for all x, x0 ∈ X, d(x, x0) < ∞
• compact set
X ⊂ RL compact if X closed and bounded
• boundary of X
boundary point x ∈ X: every N(x) contains x0 ∈ X
and x0 6∈ X
boundary of X = set of all boundary points
example: X = (0, 1], boundary points = {0, 1}
• closure of set X: X
X ≡ X∪ boundary of X
→X⊆X
→ X = X iff X closed
c Ronald Wendner
GE-Math-8
v2.1
• interior X = biggest open set contained in X
→ interior X ⊆ X
→ interior X = X iff X open
→ boundary of X = X\ interior X
• convex set
x, x0 ∈ X and λ ∈ [0, 1]
convex combination x00 ≡ λ x + (1 − λ)x0
convex set: x00 ∈ X for all x, x0 ∈ X and λ ∈ [0, 1]
examples (→ class)
if X, X 0 convex, then:
X, X, X ∩ X 0, X + X 0 convex
5
Brower’s Fixed-Point Theorem
Theorem 3 (Brouwer) Suppose that X ⊂ RL is nonempty,
compact, and convex. If f : X → X is a continuous function
from X to itself, then f (.) has a fixed point; i.e., there is an
x ∈ X such that: x = f (x).
in which way may the theorem fail to hold if:
→
→
→
→
f (x) is not continuous
X is not closed
X is not bounded
X is not convex
c Ronald Wendner
GE-Math-9
v2.1
6
Separation Theorems
Consider p ∈ RL
• hyperplane H(p, k) ≡ {x ∈ RL | p · x = k}
→ budget “line”
(p = prices, x = consumption bundle, k = wealth)
→ isoprofit “line” (k = profit)
Lemma 1 Consider K ⊂ RL nonempty, closed, convex, and
z 6∈ K, z ∈ RL. Then ∃y ∈ K and p ∈ RL\{0}: p · z < k =
p · y ≤ p · x for all x ∈ K.
→ exists a hyperplane separating z from K and
bounding for K (K on one side of H)
Theorem 4 (Bounding H Theorem, Minkowski) .
Consider K ⊂ RL, convex, and z ∈ boundary K. Then there
exists a bounding hyperplane through z.
→ there exists a bounding hyperplane for K
→ illustrate theorem by figure
c Ronald Wendner
GE-Math-10
v2.1
Theorem 5 (Separating H Theorem) .
Consider A, B ⊂ RL, nonempty, convex, disjoint: A ∩ B = ∅.
Then, there exists p ∈ RL\{0}: p·a ≥ p·b, for all a ∈ A, b ∈ B.
→ there exists a separating hyperplane for A and B
→ illustrate theorem by figure
→ results needed for welfare analysis
(e.g., 2nd FUN theorem)
c Ronald Wendner
GE-Math-11
v2.1