Lecture 2
ECON30020: Mathematical Economics
S. R. Williams
UniMelb
March 1, 2024
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Today’s Class
topology
the real line R and Euclidean space Rn
closed and open sets, compactness
dot (scalar) product
functions
level sets
examples in economics
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The Real Line
a depiction of the set R of real numbers
open interval (a, b) = {x ∈ R |a < x < b}
closed interval [c , d ] = {x ∈ R |c ≤ x ≤ b}
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Euclidean Space
Rn = {x = (x1, x2, . . . , xn) |xi ∈ R for 1 ≤ i ≤ n}
x denotes a vector, a point, or an n-tuple
“bold” used for vectors
Euclidean distance: for x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn),
the distance ∥x− y∥ between y and x is
∥x− y∥ =
√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
length of a vector x (or distance from 0 = (0, 0, . . . , 0)) is
∥x∥ =
√
x21 + x
2
2 + . . .+ x2n
on the real line R,
∥x − y∥ = |x − y | (absolute value)
∥x∥ = |x |
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Open Balls
topos: “place” (Greek)
generalizing open and closed intervals to Euclidean space Rn
open ball of radius ε at the point x0 ∈ Rn : Bε(x0)
all points whose distance from x0 is less than ε
expresses “nearness” to the center x0
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Closed Balls
closed ball of radius ε at the point x0 ∈ Rn: all points whose
distance from x0 is less than or equal to ε
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Open and Closed Sets
X ⊂ Rn is open if, for every x0 ∈ X , there exists an ε (x0) > 0 such
that Bε(x0)
(
x0
) ⊂ X
expresses the ability to move around any point x0 without leaving the
set X
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Open and Closed Sets (Cont.)
Y ⊂ Rn is closed if its complement Rn\Y is an open set
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Open and Closed Sets in the Real Line
every open interval is an open set, every closed interval is a closed set
an open set need not be an open interval, and a closed set need not
be a closed interval
can be disjoint, separated sets
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Bounded Set in Euclidean Space
A set X ⊂ Rn is bounded if there exists an open ball Bε (x0) that
contains it
ε is not required here to be small
a set is bounded if and only if there is an upper bound on how far apart
any two of its elements can be
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Boundary Point of a Set
x ∈ Rn is a boundary point of the set X ⊂ Rn if for every δ > 0,
each of the sets
Bδ(x) ∩ X ,Bδ(x) ∩Rn\X
is non-empty.
every ball Bδ(x) centered at x includes points in X and points outside
of X (in Rn\X )
example: the boundary points of the ball Bε(x0) consists of all points
that are exactly ε in distance from x0
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A Set is Closed
It Contains All of its Boundary Points
Suppose X ⊂ Rn is closed
show that every boundary point of X lies in X
proof by contradiction: suppose x∗ is a boundary point of X and
x∗ /∈ X
because X is closed, its complement Rn\X is open
x∗ ∈ Rn\X
there exists Bε(x∗) ⊂ Rn\X
this contradicts the assumption that x∗ is a boundary point of X
conclude: every boundary point of X lies in X
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A Set Contains All of its Boundary Points
It is Closed
Suppose every boundary point of X ⊂ Rn is in X
prove that X must be closed
proof by contradiction: suppose x∗ ∈ Rn\X has the property that
every Bε(x∗) ∩ X is nonempty
we also have:Bε(x∗)∩Rn\X is nonempty because x∗ ∈ Bε(x∗),Rn\X
x∗ is therefore a boundary point of X
this contradicts the fact that every boundary point of X lies in X
conclude: X is closed
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Compact
a set X is compact if it is both closed and bounded
along with continuity of a function, this condition ensures existence of
extreme values (maxima and minima) in X
this will come next week, after we discuss continuity
Weierstrass Theorem
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Dot (Scalar, or Inner) Product of Two Vectors
Let x = (x1, x2, . . . , xn) and y = (y1, y2, , . . . , yn). Then
x · y = x1y1 + x2y2 + . . .+ xnyn
start with two vectors x, y, end up with a real number x · y
∥x∥ = √x · x
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Use: Budget Constraint
let:
x = (x1, x2, . . . , xn) denote a bundle of goods, where each xi ≥ 0
p = (p1, p2, , . . . , pn) a vector of prices, where each pi > 0
I > 0 the income or wealth of a consumer.
the consumer’s budget constraint is
p1x1 + p2x2 + . . .+ pnxn = p · x ≤ I
use dot product when > 2 goods
his choice set is
C (p, I ) = {x ∈ Rn |each xi ≥ 0,p · x ≤ I }
= {x ∈ Rn+ |p · x ≤ I } .
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Interpretation
x · y = ∥x∥ · ∥y∥ · cos θ
where θ is the angle between the vectors x and y
x · y = 0 implies that the vectors x and y are perpendicular
used in interpreting gradients and directional derivatives in
multivariable calculus
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Functions
Let X and Y be sets. A function f from X to Y is a rule that assigns an
element of Y to each element of X
denoted:
y = f (x), f : X → Y
starting point: X , Y ⊂ R
a function f in this case may be given by a formula, e.g.,
f (x) = x2 − 1,
or, it may be defined in some other way e.g., f (x) = the largest integer
that is less than or equal to x
more generally: X and Y can be any sets
essential point is that a unique element of Y is specified for each
element of X
not every element of Y need be assigned
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Terminology of Functions
f : X → Y
X : the domain of the function f
Y : the range of the function f
sometimes called the codomain
f (X ): the image of the function f
f (X ) ⊂ Y is the set consisting of all elements of the range Y that are
assigned by f to some element of X
f (X ) = {y ∈ Y |y = f (x) for some x ∈ X }
for x ∈ X , f (x) is an element of Y , and f (X ) is a subset of Y
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Terminology of Functions (Cont.)
for y ∈ Y , f −1(y) ⊂ X is the inverse image (or preimage)
determined by y
the set of all elements in X such that are assigned y by f :
f −1(y) = {x ∈ X |f (x) = y }
when X ⊂ Rn and y ∈ R, an inverse image is sometimes called a
level set of f
more generally: for any subset Y ′ ⊂ Y of the range,
f −1(Y ′) =
{
x ∈ X ∣∣f (x) ∈ Y ′ }
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Examples of Functions in Economics
the consumption set for a consumer is
Rn+ = {(x1, x2, . . . , xn) ∈ Rn |each xi ≥ 0}
xi denotes a nonnegative quantity of good i
a utility function is a function
u : Rn+ → R
that represents the consumer’s preferences over bundles of goods
x ∈ Rn+
a level set u−1(k) of u determined by k ∈ R is an indifference curve
for the consumer, i.e., all bundles x ∈ Rn+ that provide the consumer
with utlity k :
u−1(k) = {x ∈ Rn+ |u(x) = k }
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Examples of Functions in Economics (Cont.)
the input set for a firm is
Rn+ = {(x1, x2, . . . , xn) ∈ Rn |each xi ≥ 0}
here, xi denotes a nonnegative quantity of input i
a production function is a function
q : Rn+ → R
that represents the maximal amount of an output that the firm can
produce using the inputs x ∈ Rn+
a level set q−1(k) of q determined by k ∈ R is an isoquant for the
firm, i.e., all bundles x ∈ Rn+ of inputs that the firm could use to
produce the output k
q−1(k) = {x ∈ Rn+ |q(x) = k }
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