Introduction to Mathematical Economics

TA Session

David Ihekereleome Okorie

October 1st, 2018.

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 1 / 26
Outline

1 Rules

2 Session Materials

3 Sequences and Sets

4 Bounded Set

5 Existence of Optimal Solution

6 Derivatives

7 Partial Derivatives

8 Total Derivatives

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 2 / 26
 Rules

Outline

1 Rules

2 Session Materials

3 Sequences and Sets

4 Bounded Set

5 Existence of Optimal Solution

6 Derivatives

7 Partial Derivatives

8 Total Derivatives

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 3 / 26
 Rules

...

You must ...

not use your phone(s) during the classes.
ask ’any’ question(s) bothering you.
discuss any concerns you have about the course with me or
Prof. Ziyan

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 4 / 26
 Session Materials

Outline

1 Rules

2 Session Materials

3 Sequences and Sets

4 Bounded Set

5 Existence of Optimal Solution

6 Derivatives

7 Partial Derivatives

8 Total Derivatives

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 5 / 26
 Session Materials

...

Session website
1. Visit www.sweetdavo.weebly.com

2. Navigate to teaching materials page

3. Click to download

My office hour
Office hour holds every Monday, 11:50am - 12:50pm.

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 6 / 26
 Sequences and Sets

Outline

1 Rules

2 Session Materials

3 Sequences and Sets

4 Bounded Set

5 Existence of Optimal Solution

6 Derivatives

7 Partial Derivatives

8 Total Derivatives

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 7 / 26
 Sequences and Sets

Sequence

A sequence in Rn is an infinite set of points xk where xk  Rn for each

integer k = {1, 2, 3, 4, ...}
Examples
{1,0,1,0,0,1, ...}
{1,2,3,4,5, ...}
1
xk = 1 − k for k = 1, 2, 3, ...

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 8 / 26
 Sequences and Sets

Remarks:
1. A sequence converges to a limit. That is to say d(xk , x) → 0 as
k → ∞. The limit here is what?

2. A sequence is monotone increasing if xk+1 ≥ xk and

monotone decreasing if xk+1 ≤ xk

3. A sequence is bounded by a and b if ∃ a, b  R such that

a ≤ xk ≤ b ∀ k.

4. Every monotone and bounded sequence converges.

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 9 / 26
 Sequences and Sets

Sets
An open ball or neighbourhood with centre x and radius r is defined as
B(x,r) = {y  Rn | d(x,y) < r}.
Open and closed sets
1. A set S ⊆ Rn is open iff ∀ x  S, ∃ any radius, r > 0, such that
B(x,r)  S. Hence, for each x  S, there is an open ball around x that is
contained entirely in S

2. A set S ⊆ Rn is closed iff ∀ x  S and xk → x, wherein

x  S. Hence, a closed set contains its limit points.

Egs. Sketch the first two:

1. {(x1 , x2 )  Rn | a1 < x1 < b1 , a2 < x2 < b2 }
2. {(x1 , x2 )  Rn | a1 ≤ x1 ≤ b1 , a2 ≤ x2 ≤ b2 }
3. Which is open and which is closed?
4. is [2,8) open or closed ?
David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 10 / 26
 Bounded Set

Outline

1 Rules

2 Session Materials

3 Sequences and Sets

4 Bounded Set

5 Existence of Optimal Solution

6 Derivatives

7 Partial Derivatives

8 Total Derivatives

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 11 / 26
 Bounded Set

Bounded Set
A set S ⊂ Rn is bounded if ∃ r > 0 such that S ⊂ B(0,r) is defined/exist.
Hence, the ball completely contains S for any finite radius.

Egs. Find the radius,r, that makes the following bounded sets
1. S = [0, 2]
2. S = [2, 5]
3. S = [−2, 2]
4. S = {0, 10, 20, ...}

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 12 / 26
 Bounded Set

Upper Bound
Given A ⊂ R, u ⊂ R is an upper bound of A if u ≥ a ∀ A. U(A) is the
set for all upper bonds of A.

Lower Bound
Given A ⊂ R, l ⊂ R is a lower bound of A if l ≤ a ∀ A. L(A) is the set
for all lower bonds of A.

Supremum
This is the least upper bound. sup(A) ≤ u ∀ u  U(A). sup(A) is unique.
sup(A) can be ∞ (not well defined) if A is not bounded above

Infimum
This is the highest lower bound. inf(A) ≥ l ∀ l  L(A). sup(A) is unique.
sup(A) can be -∞ (not well defined) if A is not bounded above

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 13 / 26
 Existence of Optimal Solution

Outline

1 Rules

2 Session Materials

3 Sequences and Sets

4 Bounded Set

5 Existence of Optimal Solution

6 Derivatives

7 Partial Derivatives

8 Total Derivatives

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 14 / 26
 Existence of Optimal Solution

Compact Set
A set S ⊂ Rn is compact if every sequence in S contains a convergent
subsequence. That is, set S ⊂ Rn is compact iff it is closed and bounded.

Weierstrass Theorem:
Let D ⊂ R be compact and let f:D → R is a continuous function on
D, then ∃ points z1 and z2 in D such that f (z1 ) ≥ f (x) ≥ f (z2 ), x  D.
Hence, f attains a maximum and a minimum.

What are the maximum and minimum points?

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 15 / 26
 Existence of Optimal Solution

Examples:
Determine the Maximum, Minimum, Infimum,and Supremum of the
following:
1).
D = (−1, 1), f (x) = x2
2). (
x, if x = n1 , n = 1, 2, 3, ...,
D = [0, 1], f (x) =
1, otherwise
3).
D = R, f (x) = −|x|
4).
D = R, f (x) = |x|
5).
max U (C)
C
Given C = {C|0 ≤ PC ≤ I}, I,P > 0.
What assumption do we need to guarantee that a solution exists?
David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 16 / 26
 Derivatives

Outline

1 Rules

2 Session Materials

3 Sequences and Sets

4 Bounded Set

5 Existence of Optimal Solution

6 Derivatives

7 Partial Derivatives

8 Total Derivatives

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 17 / 26
 Derivatives

Differentiability and Continuity

Domain
Domain of f is the set of numbers x at which f(x) is defined.
What is Range?

Differentiable
A differentiable function f is differentiable at every point x0 in its domain
D. i.e. the curve of f is smooth.

Continuous
A continuous function f, for any sequence {xn } which converges to x0 in
the domain, D, f (xn ) converges to f (x0 ). i.e. there are no breaks in the
graph.

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 18 / 26
 Derivatives

Continuously Differentiable function (C 1 )

A function f is continuously differentiable if f 0 (x) is continuous.

Twice Continuously Differentiable function (C 2 )

A function f is twice continuously differentiable if f 00 (x) is continuous.

Derivative
dy f (x + h) − f (x)
f 0 (x) = = lim
dx h→0 h

Second Derivative
d2 y f 0 (x + h) − f 0 (x)
f 00 (x) = = lim
dx2 h→0 h

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 19 / 26
 Derivatives

Some Rules of Differentiation

D(xk ) = d(xk ) = (xk )0 = kxk−1 , called Power Rule
D(α) = 0 why?
(f ± g)0(x) = f 0(x) ± g0(x), called Sum & Difference Rule
(f • g)0(x) = f 0(x)g(x) + f (x)g0(x), called Product Rule
f 0(x)g(x)−f (x)g0(x)
( fg )0(x) = (g(x))2
, called Quotient Rule
Df (g(x)) = f 0(g(x))g0(x), called Chain Rule
1
D(lna (x)) = xlna , called Logarithmic Rule

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 20 / 26
 Derivatives

Examples

f (w) = 7w4 − 8w2 + 9w find f 0(w),f 0(1),f 00(w), and f 00(w = 2)

f (x) = (1 − x3 )5 , find f 0

1
k = ( x−1
x+3 ) find
3
dk
dx

y = (x − 3)(x2 + 8)3 find y0

V = (mpk )(r − p) find Vp

Y = Ak α lβ find Yk ,Yl ,Ykl

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 21 / 26
 Partial Derivatives

Outline

1 Rules

2 Session Materials

3 Sequences and Sets

4 Bounded Set

5 Existence of Optimal Solution

6 Derivatives

7 Partial Derivatives

8 Total Derivatives

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 22 / 26
 Partial Derivatives

Consider f (x) = f (x1 , x2 , ..., xk , ..., xn ) where xi can vary without af-
fecting others. i.e. xi changes by ∆xi while other x’s remain unchanged,
y will change by ∆y.
Definition
δf f (x1 , x2 , ..., xk + h, ..., xn ) − f (x1 , x2 , ..., xk , ..., xn )
= lim
δxk h→0 h

Examples
f (x, y) = 8xy − x3 y + xy 5 find f1 , f2 , f21 , and f1 (2, 4)

g(k, m) = 75k 4 find f1 and fm

g(s + q) = α(s + q)β − ms1−α + pq r+1 , find g0, gs and gq

What did you observe?

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 23 / 26
 Total Derivatives

Outline

1 Rules

2 Session Materials

3 Sequences and Sets

4 Bounded Set

5 Existence of Optimal Solution

6 Derivatives

7 Partial Derivatives

8 Total Derivatives

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 24 / 26
 Total Derivatives

Consider f (x) = f (x1 , x2 , ..., xk , ..., xn ) where all xi change

simultaneously . i.e. all xi change by ∆xi , y will change by ∆y, total
change (dy).
Definition
df f (xi + ∆) − f (xi )
= lim =
dx ∆→0 ∆

f (x1 + ∆x1 , x2 + ∆x2 , ..., xk + ∆xk , ..., xn + ∆xn ) − f (x1 , x2 , ..., xk , ..., xn )
lim
∆→0 ∆
where ∆ is a vector.

Therefore
δF δF δF δF
dF = dx1 + dx2 + dxk + ... + dxn
δx1 δx2 δxk δxn

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 25 / 26
 Total Derivatives

Examples
m = x6 lny, find dm

y = 5p3 qr + rpq − 9r2 , find dy

...

Q&A

David Ihekereleome Okorie (XMU) Intro to Math Econ October 1st, 2018. 26 / 26