Lecture 1
Quantitative Methods in Economics and Business
Mikhail Anufriev
based on previous slides of Benjamin Balzer
University of Technology Sydney
Spring 2023
Why math?
Your major is economics/business. Why do you have to take a math
class?
The language of economics/business is mathematics.
To answer questions like How does a tax affect the amount and the
price of a good that is traded on a market, economists build a model.
Definition
Modelling: The creation of a piece of mathematical theory which represents
(a simplification of) some aspects of practical economics.
We will see examples of economic models during the course.
Aim of this course: Introduce you to the basic mathematics used in
economics
Learning goals
At the end of the course you should understand and be confident to
apply the basic concepts of mathematics that are used in economics
Mathematics are a tool used to answer economic questions.
If you are confident with the language (math), you can focus on the
content of the model(the economics)(⇒ you get the most out of your
studies)
Having a good background in mathematics is an advantage in most of
the other courses you will encounter during your studies
⇒ It is a valuable investment. You will profit during your future
studies!
How to get the most out of this Course?
Read the relevant book chapters before the lecture
Try to solve the tutorial exercises before going there
Practice (often times complicated things become fairly approachable
once you practiced them a lot)
there are a lot of exercises the book (sample solutions are at the back).
A lot of other material as U:Pass, Khans Academy,...
This week: Review of basic concepts
You should know how to
Add, subtract, multiply and divide
Multiply out brackets
Simplify and factorize algebraic expressions
Evaluate algebraic expressions numerically
Solve equations and linear inequalities
Understand the graphical representation of linear equations
Tutorial 0 (on Canvas) is about these concepts. Solve the exercises . If
you have problems read chapters 1.1,1.2,1.3 and the end of chapter 1
or watch the youtube videos linked under page “Week 1: Before
Class”
Tutorial 1 will give you an additional training session
Today: Short review of notation and basic concepts we will
use in the course
Basic knowledge of algebra (Chapter: 1.1, 1.2)
Graphs of linear equations (Chapter: 1.3)
Some formal mathematics (End of Chapter 1)
Numbers and Variables
Numbers and Variables
Real numbers (R) can be represented on the real line:
−∞
−3.5
−2
0
1 1.859... 4
∞
Numbers and Variables
Real numbers (R) can be represented on the real line:
−∞
−3.5
−2
0
1 1.859... 4
∞
Letters (e.g., P, Q, a, b , x, y) are called variables.
Variables are placeholders for numbers.
In this course, if not explicitly stated otherwise, every variable is a
real number.
Endogenous vs Exogenous Variables
Next lecture we will see the difference between exogenous variables
(parameters or coefficients) and endogenous variables within an
economic context
Put roughly: endogenous variables are ”unknowns” (which are
determined within a model) and exogenous variables are constants
(just think of them as a fancy version of a given real number)
Endogenous vs Exogenous Variables
Next lecture we will see the difference between exogenous variables
(parameters or coefficients) and endogenous variables within an
economic context
Put roughly: endogenous variables are ”unknowns” (which are
determined within a model) and exogenous variables are constants
(just think of them as a fancy version of a given real number)
Example 1
Problem: Find x that solves 1 + x = 1.
Solution: x = 0
Endogenous vs Exogenous Variables
Next lecture we will see the difference between exogenous variables
(parameters or coefficients) and endogenous variables within an
economic context
Put roughly: endogenous variables are ”unknowns” (which are
determined within a model) and exogenous variables are constants
(just think of them as a fancy version of a given real number)
Example 1
Problem: Find x that solves 1 + x = 1.
Solution: x = 0
Example 2 :
Problem: Let a be a real number. Find x that solves a + x = 1.
Solution: x = 1 − a.
Endogenous vs Exogenous Variables
Next lecture we will see the difference between exogenous variables
(parameters or coefficients) and endogenous variables within an
economic context
Put roughly: endogenous variables are ”unknowns” (which are
determined within a model) and exogenous variables are constants
(just think of them as a fancy version of a given real number)
Example 1
Problem: Find x that solves 1 + x = 1.
Solution: x = 0
Example 2 :
Problem: Let a be a real number. Find x that solves a + x = 1.
Solution: x = 1 − a.
In both examples x is an endogenous variable, because it is the
solution to an equation. a is an exogenous variable, because we take it
as given. Typically the value of an endogenous variable, that is, the
value of the unknown, depends on the exogenous variables.
Algebra
Basic Knowledge of Algebra
Multiplication: a × b = ab and a × (−b) = −ab
Basic Knowledge of Algebra
Multiplication: a × b = ab and a × (−b) = −ab
If a is a positive number and b is a negative number then ab is a
negative number
Basic Knowledge of Algebra
Multiplication: a × b = ab and a × (−b) = −ab
If a is a positive number and b is a negative number then ab is a
negative number
Notation for multiplying a k-times by itself: a × a × ...× = ak , k is
called index.
Basic Knowledge of Algebra
Multiplication: a × b = ab and a × (−b) = −ab
If a is a positive number and b is a negative number then ab is a
negative number
Notation for multiplying a k-times by itself: a × a × ...× = ak , k is
called index.
Distributive law: a(b + c) = ab + ac
Basic Knowledge of Algebra
Multiplication: a × b = ab and a × (−b) = −ab
If a is a positive number and b is a negative number then ab is a
negative number
Notation for multiplying a k-times by itself: a × a × ...× = ak , k is
called index.
Distributive law: a(b + c) = ab + ac
Factorize [Reverse operation of distributive law]. Example:
6a − 3a2 = 3 × 2 × a − 3 × a × a = 3a(2 − a)
Basic Knowledge of Algebra
Multiplication: a × b = ab and a × (−b) = −ab
If a is a positive number and b is a negative number then ab is a
negative number
Notation for multiplying a k-times by itself: a × a × ...× = ak , k is
called index.
Distributive law: a(b + c) = ab + ac
Factorize [Reverse operation of distributive law]. Example:
6a − 3a2 = 3 × 2 × a − 3 × a × a = 3a(2 − a)
Difference of Square Formula:
(a + b)(a − b) = a2 − b 2
Basic Knowledge of Algebra
Multiplication: a × b = ab and a × (−b) = −ab
If a is a positive number and b is a negative number then ab is a
negative number
Notation for multiplying a k-times by itself: a × a × ...× = ak , k is
called index.
Distributive law: a(b + c) = ab + ac
Factorize [Reverse operation of distributive law]. Example:
6a − 3a2 = 3 × 2 × a − 3 × a × a = 3a(2 − a)
Difference of Square Formula:
(a + b)(a − b) = a2 − b 2
Rule: Brackets Indixes Devision Multiplication Addition Subtraction
Example:
2b(b×2)−2(b−1+1)2 = 2b(2b)−2(b)2 = 2b(2b)−2b 2 = 4b 2 −2b 2 = 2b 2
Fractions
Fraction ba stands for a divided by b. a is called numerator, and b is
called denominator.
Example:
1
2
= 0.5,
a
a
= 1,
a
1
= a.
Rules
Multiplication
a
c
a×c
ac
× =
=
b d
b×d
bd
Fractions
Fraction ba stands for a divided by b. a is called numerator, and b is
called denominator.
Example:
1
2
= 0.5,
a
a
= 1,
a
1
= a.
Rules
Multiplication
Division
a
c
a×c
ac
× =
=
b d
b×d
bd
a c
a
d
/ = ×
b d
b
c
Fractions
Fraction ba stands for a divided by b. a is called numerator, and b is
called denominator.
Example:
1
2
= 0.5,
a
a
= 1,
a
1
= a.
Rules
Multiplication
Division
a
c
a×c
ac
× =
=
b d
b×d
bd
a c
a
d
/ = ×
b d
b
c
Adding
a
c
ad
cb
ad + cb
+ =
+
=
b d
bd
db
db
Fractions
Fraction ba stands for a divided by b. a is called numerator, and b is
called denominator.
Example:
1
2
= 0.5,
a
a
= 1,
a
1
= a.
Rules
Multiplication
Division
a
c
a×c
ac
× =
=
b d
b×d
bd
a c
a
d
/ = ×
b d
b
c
Adding
a
c
ad
cb
ad + cb
+ =
+
=
b d
bd
db
db
Subtracting
a
c
ad
cb
ad − cb
− =
−
=
b d
bd
db
db
Equations
Expressions like a = b + c are called equations.
Equations
Expressions like a = b + c are called equations.
Rules:
equality does not change if the same quantity is added to (or subtracted
from) both sides
Equations
Expressions like a = b + c are called equations.
Rules:
equality does not change if the same quantity is added to (or subtracted
from) both sides
equality does not change if both sides are multiplied by the same
quantity (or divided by the same non-zero quantity)
Equations
Expressions like a = b + c are called equations.
Rules:
equality does not change if the same quantity is added to (or subtracted
from) both sides
equality does not change if both sides are multiplied by the same
quantity (or divided by the same non-zero quantity)
division of zero is forbidden
if the product of two numbers is zero, then at least one of them must be
zero.
Notation/ Logical Operators
⇒ and ⇔
Equation 1 implies equation 2 if we can derive equation 2 from
equation 1 by applying algebra. We write equation 1 ⇒ equation 2.
⇒ and ⇔
Equation 1 implies equation 2 if we can derive equation 2 from
equation 1 by applying algebra. We write equation 1 ⇒ equation 2.
If equation 1 ⇒ equation 2 and equation 2 ⇒ equation 1, then
equation 1 is equivalent to equation 2 and we write equation 1 ⇔
equation 2.
⇒ and ⇔
Equation 1 implies equation 2 if we can derive equation 2 from
equation 1 by applying algebra. We write equation 1 ⇒ equation 2.
If equation 1 ⇒ equation 2 and equation 2 ⇒ equation 1, then
equation 1 is equivalent to equation 2 and we write equation 1 ⇔
equation 2.
Example: a = b ⇔ a + 1 = b + 1. Why?
1
a=b
|+1
⇒a+1=b+1
2
a+1=b+1
|−1
⇒a=b
Example: a = b ⇒
a2
=
b2 ,
but a2 = b 2
⇒
|{z}
not necessarily
tutorial 1)
a = b. (see
⇒ and ⇔: More general
Statement A implies statement B if we can derive B from A by using
mathematics (often logic). We write A ⇒ B
If A ⇒ B and B ⇒ A, then B is equivalent to A and we write A ⇔ B
we write (in words) A if and only if (iff) B.
⇒ and ⇔: More general
Statement A implies statement B if we can derive B from A by using
mathematics (often logic). We write A ⇒ B
If A ⇒ B and B ⇒ A, then B is equivalent to A and we write A ⇔ B
we write (in words) A if and only if (iff) B.
Example:
Statement A: Today is Australia Day
Statement B: Today is the 26th January
A ⇒ B and B ⇒ A. Thus A ⇔ B
⇒ and ⇔: More general
Statement A implies statement B if we can derive B from A by using
mathematics (often logic). We write A ⇒ B
If A ⇒ B and B ⇒ A, then B is equivalent to A and we write A ⇔ B
we write (in words) A if and only if (iff) B.
Example:
Statement A: Today is Australia Day
Statement B: Today is the 26th January
A ⇒ B and B ⇒ A. Thus A ⇔ B
Example:
Statement A: Today is a day in January
Statement B: Today is Australia Day
B ⇒ A, but A does not imply B.
Equations and Identities
Equation:
An equality that is true for particular values of a variable
Identity:
An equality that is true for all values of a variable
Equations and Identities
Equation:
An equality that is true for particular values of a variable
Identity:
An equality that is true for all values of a variable
Example:
a + 1 = 2 is an equation, because the equality is only true for a = 1.
Equations and Identities
Equation:
An equality that is true for particular values of a variable
Identity:
An equality that is true for all values of a variable
Example:
a + 1 = 2 is an equation, because the equality is only true for a = 1.
2a = a + a is an identity, because the equality is true for all values of a.
Equations and Identities
Equation:
An equality that is true for particular values of a variable
Identity:
An equality that is true for all values of a variable
Example:
a + 1 = 2 is an equation, because the equality is only true for a = 1.
2a = a + a is an identity, because the equality is true for all values of a.
Sometimes people use the sign ≡ for an identity (I will use ≡ instead of
= whenever I think it helps to clarify the exposition)
Equations and Identities
Equation:
An equality that is true for particular values of a variable
Identity:
An equality that is true for all values of a variable
Example:
a + 1 = 2 is an equation, because the equality is only true for a = 1.
2a = a + a is an identity, because the equality is true for all values of a.
Sometimes people use the sign ≡ for an identity (I will use ≡ instead of
= whenever I think it helps to clarify the exposition)
The sign ≡ is often useful when it comes to definitions. For example,
average costs, say AC (Q), are defined as total costs, C (Q), divided by
the total quantity produced, Q. When defining the average costs
mathematically, we might write AC (Q) ≡ C (Q)
Q . When using the = sign
instead, we would wonder whether we are looking for the Q that satisfies
the equality AC (Q) = C (Q)
Q , rather than observing that this equality
holds for all Q by definition.
Inequalities
Inequalities
(weak) inequalities: a ≥ b: the term on the left hand side is weakly
larger than the term on the right hand side.
Example: 2 ≥ 1, 1 ≥ 1
(strict) Inequality : a > b, the term on the left is strictly larger than
the term on the right hand side.
Example: 2 > 1
If a ≥ b, and b ≥ a, then a = b.
Inequalities
(weak) inequalities: a ≥ b: the term on the left hand side is weakly
larger than the term on the right hand side.
Example: 2 ≥ 1, 1 ≥ 1
(strict) Inequality : a > b, the term on the left is strictly larger than
the term on the right hand side.
Example: 2 > 1
If a ≥ b, and b ≥ a, then a = b.
Rules:
If a ≥ b, then a − c ≥ b − c.
Inequalities
(weak) inequalities: a ≥ b: the term on the left hand side is weakly
larger than the term on the right hand side.
Example: 2 ≥ 1, 1 ≥ 1
(strict) Inequality : a > b, the term on the left is strictly larger than
the term on the right hand side.
Example: 2 > 1
If a ≥ b, and b ≥ a, then a = b.
Rules:
If a ≥ b, then a − c ≥ b − c.
If a ≥ b, then a + c ≥ b + c.
Inequalities
(weak) inequalities: a ≥ b: the term on the left hand side is weakly
larger than the term on the right hand side.
Example: 2 ≥ 1, 1 ≥ 1
(strict) Inequality : a > b, the term on the left is strictly larger than
the term on the right hand side.
Example: 2 > 1
If a ≥ b, and b ≥ a, then a = b.
Rules:
If a ≥ b, then a − c ≥ b − c.
If a ≥ b, then a + c ≥ b + c.
Let c < 0. If a ≥ b, then ac ≤ bc
Inequalities
(weak) inequalities: a ≥ b: the term on the left hand side is weakly
larger than the term on the right hand side.
Example: 2 ≥ 1, 1 ≥ 1
(strict) Inequality : a > b, the term on the left is strictly larger than
the term on the right hand side.
Example: 2 > 1
If a ≥ b, and b ≥ a, then a = b.
Rules:
If a ≥ b, then a − c ≥ b − c.
If a ≥ b, then a + c ≥ b + c.
Let c < 0. If a ≥ b, then ac ≤ bc
Let d > 0. If a ≥ b, then ad ≥ bd
Intervals
Intervals (Formal mathematics)
The symbol [a, b] stands for the set of all numbers that are weakly
larger than a and weakly lower than b.
[a, b] is called (closed) interval
x ∈ [a, b] means: The variable x is included in [a, b]. This is true for
every value of x such that a ≤ x ≤ b
Example: [1, 2] are all real numbers between 1 and 2. Thus, the
following is correct: 1.5 ∈ [1, 2]
Intervals (Formal mathematics)
The symbol [a, b] stands for the set of all numbers that are weakly
larger than a and weakly lower than b.
[a, b] is called (closed) interval
x ∈ [a, b] means: The variable x is included in [a, b]. This is true for
every value of x such that a ≤ x ≤ b
Example: [1, 2] are all real numbers between 1 and 2. Thus, the
following is correct: 1.5 ∈ [1, 2]
Open interval (a, b). x ∈ (a, b) ⇔ a < x < b
Example: 1.1 ∈ (1, 2) but 1 ∈
/ (1, 2)
Intervals (Formal mathematics)
The symbol [a, b] stands for the set of all numbers that are weakly
larger than a and weakly lower than b.
[a, b] is called (closed) interval
x ∈ [a, b] means: The variable x is included in [a, b]. This is true for
every value of x such that a ≤ x ≤ b
Example: [1, 2] are all real numbers between 1 and 2. Thus, the
following is correct: 1.5 ∈ [1, 2]
Open interval (a, b). x ∈ (a, b) ⇔ a < x < b
Example: 1.1 ∈ (1, 2) but 1 ∈
/ (1, 2)
Half-open interval [a, b). x ∈ [a, b) ⇔ a ≤ x < b
Intervals (Formal mathematics)
The symbol [a, b] stands for the set of all numbers that are weakly
larger than a and weakly lower than b.
[a, b] is called (closed) interval
x ∈ [a, b] means: The variable x is included in [a, b]. This is true for
every value of x such that a ≤ x ≤ b
Example: [1, 2] are all real numbers between 1 and 2. Thus, the
following is correct: 1.5 ∈ [1, 2]
Open interval (a, b). x ∈ (a, b) ⇔ a < x < b
Example: 1.1 ∈ (1, 2) but 1 ∈
/ (1, 2)
Half-open interval [a, b). x ∈ [a, b) ⇔ a ≤ x < b
Half-open interval (a, b]. x ∈ (a, b] ⇔ a < x ≤ b
Intervals (Formal mathematics)
The symbol [a, b] stands for the set of all numbers that are weakly
larger than a and weakly lower than b.
[a, b] is called (closed) interval
x ∈ [a, b] means: The variable x is included in [a, b]. This is true for
every value of x such that a ≤ x ≤ b
Example: [1, 2] are all real numbers between 1 and 2. Thus, the
following is correct: 1.5 ∈ [1, 2]
Open interval (a, b). x ∈ (a, b) ⇔ a < x < b
Example: 1.1 ∈ (1, 2) but 1 ∈
/ (1, 2)
Half-open interval [a, b). x ∈ [a, b) ⇔ a ≤ x < b
Half-open interval (a, b]. x ∈ (a, b] ⇔ a < x ≤ b
Prominent intervals:
R = (−∞, ∞),
R+ = (0, ∞),
R− = (−∞, 0)
Graphs of Linear Equations
Graphs of Linear Equations (1.3)
Linear equation:
dx + ey = f
(1)
Suppose we are interested in all values of x and y that satisfy equation
(1) for given values of d,e,f
We call d,e,f coefficients (or parameters).
Graphs of Linear Equations (1.3)
Linear equation:
dx + ey = f
(1)
Suppose we are interested in all values of x and y that satisfy equation
(1) for given values of d,e,f
We call d,e,f coefficients (or parameters).
We can graph the equation in a coordinate system.
1
If e ̸= 0, we isolate y , that is,
dx + ey = f
2
⇔ ey = f − dx
d
f
⇔y = − x
(2)
e
e
For every x on the x-axis we can use (2) to calculate y .
We can use the graph to
Find solutions of simultaneous linear equations (or inequalities) (see
tutorial 0,1 and the beginning of the next lecture)
Recommendation
If you want to be on top of things right from the start, then use this week
to:
1
Solve Tutorial 0. If you have problems read chapters 1.1,1.2,1.3 and
the end of chapter 1 or watch the youtube videos which are linked
under “Week 1: Before Class” on Canvas
2
Try to solve Tutorial 1 ( which will be discussed in the tutorial sessions
next week)
3
( Read chapters 1.4,1.5,1.6 to be well prepared for lecture 1 (next
week) )