30C00300 Mathematical Methods for Economists (6 cr)
2) Euclidean spaces and elements of set theory
Simon & Blume chapters: 10, 11, A1.1
Slides originally by: Timo Kuosmanen
Slides amended by: Anna Lukkarinen
Lecture held by: Anna Lukkarinen
1
Presenting sales figures verbally
• On Monday, revenue from pens was 50, on
Tuesday 33, on Wednesday 14, on Thursday 68,
and on Friday 17
• Revenue from erasers was 1 on Monday, 2 on
Tuesday, zero on Wednesday, 5 on Thursday,
and 4 on Friday
• Sharpeners generated revenues of 10 on
Monday, and 6 every following day until Friday
Presenting sales figures
in matrix form
Mo Tu We
Th Fr
Outline
1. Euclidean space
2. Vectors
3. Inner product
4. Linear dependence
5. Lines and hyperplanes
6. Sets: notations and operations
4
Who is this?
5
Euclidean space
R1 = R is the set of real numbers (i.e., the real line)
R2 is called the Euclidean plane.
Rn is the Euclidean n-space (sometimes called Cartesian space or
simply n-space)
• Rn is the space of all n-tuples of real numbers (x1, x2, ..., xn).
• Elements of Rn are called n-vectors.
6
Geometric interpretation
y
x
y
z
x
• The -plane represents the
Euclidean 2-space
• Three mutually orthogonal
coordinate lines represent
Euclidean 3-space
• Similarly, Euclidean n-space
is denoted by
(no
geometric interpretation)
Outline
1. Euclidean space
2. Vectors
3. Inner product
4. Linear dependence
5. Lines and hyperplanes
6. Sets: notations and operations
8
Vectors: Basic concepts
Component or
coordinate
x has dimension n
Example: Commodity and price vectors
• Consider n different commodities
• During a certain time period, a person
consumes c1 units of commodity 1, c2 units
of commodity 2, …, and cn units of
commodity n
• The unit price of commodity 1 is p1, the unit
price of commodity 2 is p2, …, and the unit
price of commodity n is pn
Example: Commodity and price vectors
Commodity vector c
=
Price vector p
=
Vectors in Euclidean space
Two interpretations of n-dimensional vectors:
• Location: vector x is a point in Rn .
• Displacement: vector x represents movement from the origin 0 to
point x.
x2
x2
x
x
x1
x1
12
Vectors in Euclidean space
y
( ,
• A vector has two
independent
properties:
a) Length
b) Direction
)
x
Pythagorean theorem…
c
a
b
…can you show that it is true?
c
b
a
…can you show that it is true?
…can you show that it is true?
Length of a vector
The length of vector
=
is defined as the scalar (real number)
=
+
+
Basic vector operations: Addition
a
b
a+b
b
a
Example: Commodity and price vectors
If two persons both buy the commodity vector
, then + represents the total commodity
vector bought by the two persons
+ =
+
=
Basic vector operations: Subtraction
-b
a-b
a
-b
a-b
b
What operations do these vectors represent?
a+b
a
a-b
b-a
b
2b
Example: Commodity and price vectors
If a person with commodity vector increases
consumption of all goods by 50%, then 1.5
represents his/her new commodity vector
1.5 = 1.5
=
1.
1.
1.
Economic examples of
vectors in Euclidean space
Commodity baskets: for example,
æ Food ö æ y1 ö
y=ç
÷=ç ÷
è Clothing ø è y2 ø
Factors of production (inputs):
æ Labor ö æ L ö
ç
÷ ç ÷
x = ç Capital ÷ = ç K ÷
ç Materials ÷ ç M ÷
è
ø è ø
24
Outline
1. Euclidean space
2. Vectors
3. Inner product
4. Linear dependence
5. Lines and hyperplanes
6. Sets: notations and operations
25
Inner product of vectors
Definition: let u = (u1, u2, ..., un) and v = (v1, v2, ..., vn) be two vectors
in Rn. The Euclidean inner product of u and v, written as u v, is the
number
u × v = u1v1 + u2 v2 + ... + u n vn
Note: the inner product is also referred to as the dot product or the
scalar product. [In Excel, =sumproduct(u_array, v_array)]
Note 2: the inner product should not be confused with the matrix
product or the cross product (denoted by u × v).
26
Inner product as matrix multiplication
Recall: vectors can be considered as special cases of matrices
• A row vector can be stated as a 1xn matrix
• A column vector can be stated as a nx1 matrix
Consider column vectors u = (u1 u2 ... un)´ and v = (v1 v2 ... vn)´
The inner product can be expressed using the matrix product
u ¢v = u1v1 + u2 v2 + ... + u n vn = u × v
Note: The matrix product requires that the dimensions of the two
matrices match: multiplying an m × n matrix by an n × p matrix yields
an m × p matrix.
27
Example: Inner product as matrix multiplication
=
and
=
=
…
=
+
+
=
Example: Commodity and price vectors
If a person buys commodity vector with
prices given by price vector , then the total
price of the commodity basket is
or
=
=
+
+… +
Properties of the inner product
Theorem 10.2: let u, v, and w be arbitrary vectors in Rn and let r
be an arbitrary scalar. Then
a) u v = v u,
b) u (v + w) = u v + u w,
c) u (rv) = r(u v) = (ru) v
d) u u 0,
e) u u = 0 implies u = 0,
f) (u + v) (u + v) = u u + 2(u v) + v v
30
Properties of the inner product
Theorem 10.3: Let u and v be two vectors in Rn. Let be the angle
between them. Then
v
u × v = u v cos q
where
u = u12 + u22 + ... + un2
v = v12 + v22 + ... + vn2
u
are the Euclidean norms (the lengths) of vectors u and v.
Note:
is a right angle
u v=0
31
Outline
1. Euclidean space
2. Vectors
3. Inner product
4. Linear dependence
5. Lines and hyperplanes
6. Sets: notations and operations
32
Linear combination
Consider a collection of vectors x1, x2, ..., xK.
Their linear combination is
K
y = å a i xi
i =1
= a1x1 + a 2 x 2 + ... + a K x K
æ x11 ö
æ x12 ö
æ x1K ö
ç ÷
ç ÷
ç
÷
x
x
x
= a1 ç 21 ÷ + a 2 ç 22 ÷ + ... + a K ç 2 K ÷
ç M ÷
ç M ÷
ç M ÷
ç ÷
ç ÷
ç
÷
x
x
x
è n1 ø
è n2 ø
è nK ø
where a i ¹ 0 "i
33
Example: Commodity and price vectors
If persons buy the same commodity vector and
persons buy the same commodity vector , then
+ represents the total commodity vector
bought by the + persons
+
=
+
=
+
+
+
Linear dependence
Consider a collection of vectors x1, x2, ..., xK.
1) If one of the vectors, say x1, is a linear combination of the
others, that is
K
x1 = å a i xi
i=2
then the collection of vectors is said to be linearly dependent.
2) Alternatively, the collection is linearly dependent if there exist
alpha coefficients, not all zero, such that
æ0ö
ç ÷
K
0÷
ç
a i xi = 0 =
å
çM÷
i =1
ç ÷
è0ø
35
Linear independence
If the collection of vectors x1, x2, ..., xK is not linearly dependent, then
it is said to be linearly independent.
To test for linear independence, we can check if there exist non-zero
alpha coefficients such that
K
åa x
i =1
i
i
=0
In practice, we need to solve the equation to check if
1, …, K) = (0, …, 0) is the only solution.
36
Span
The set of all possible linear combinations of K linearly independent
vectors x1, x2, ..., xK is called the span of x1, x2, ..., xK.
Note: K linearly independent vectors span the K-dimensional
Euclidean space RK.
37
Outline
1. Euclidean space
2. Vectors
3. Inner product
4. Linear dependence
5. Lines and hyperplanes
6. Sets: notations and operations
38
Lines
In R2, an equation of a straight line can be expressed as
x2 = a + bx1
Alternative forms are possible, for example,
bx1 + cx2 = a
Economic example: budget line
PF Food + PC Clothes = Total budget
39
Lines and hyperplanes
In R2, an equation of a line can be stated as
bx1 + cx2 = a
In R3, a plane can be expressed as
bx1 + cx2 + dx3 = a
In Rn, a hyperplane can be expressed as
b1x1 + b2x2 + …+ bnxn = a
or, using the inner product of vectors b and x, as
=
40
Example: Lines and hyperplanes
Economic example:
Iso-cost hyperplane
w×x = C
where
w is the vector of input prices
x is the vector of input quantities
C is the total cost
Note: hyperplanes are used in many contexts to represent the
economic constraints (e.g. budget line of the consumer) or the
objectives
41
Outline
1. Euclidean space
2. Vectors
3. Inner product
4. Linear dependence
5. Lines and hyperplanes
6. Sets: notations and operations
42
Set theory
A set is any well-specified collection of elements, with a clear criterion
for membership.
A set can be defined by simply listing its elements, for example,
A = {1,2,3}
A set can contain a finite or infinite number of elements, for example
N = {1, 2, 3, …}
The n-dimensional Euclidean space can be stated as the set
ìæ x1 ö
ü
ïç ÷
ï
ïç x12 ÷
ï
n
R =í
xi Î R "i ý
ç
÷
ïç M ÷
ï
ïè xn ø
ï
î
þ
43
Set theory
Notation: a Î A means element a is a member of set A.
aÏ A
means element a is not a member of set A.
AÍ B
means set A is a subset of set B.
AÌ B
means set A is a proper subset of set B.
Æ
denotes an empty set.
44
Set theory
Operations:
Union
A È B = { x x Î A or x Î B}
Intersect
A Ç B = { x x Î A and x Î B}
Set difference
A - B = { x x Î A and x Ï B}
45
Production possibility set
The most general representation of the firm’s production technology
is the production possibility set, formally defined as
{
T = (x, y ) Î R k++ m input vector x Î R +k can produce output vector y Î R +m
}
If production plan (x,y) is technically feasible, then (x, y ) Î T
If (x, y ) Ï T , then production plan (x,y) is infeasible.
46
Production possibility set
Some important properties of T
Free disposability:
(x, y ) Î T Þ (x + u, y - v ) Î T "u Î R k+ , v Î R +m
Constant returns to scale:
(x, y ) Î T Þ (l x, l y ) Î T "l Î R +
47
Production possibility set
Efficient subset of T (Koopmans, 1951)
Eff (T) = {(x, y ) Î T $(x% , y% ) ¹ (x, y ) : x% £ x, y% ³ y Þ (x% , y% ) Ï T}
Technical inefficiency: if (x, y) Î (T - Eff (T))
then production plan (x,y) is said to be inefficient.
48
Next time
Topic:
• Mathematical proofs
Textbook:
• Simon & Blume, Appendix A1.3
49