ECON 106
Chiang and Wainwright, Fundamental Methods of Mathematical Economics
CHAPTER 1
What is Mathematical Economics?
● approach to economic analysis using
mathematics
● economists construct precisely defined
models from which exact conclusions
can be derived with mathematical logic
and use these to make quantifiable
predictions about future economic
activity.
Why use math in economics?
1. Math is a precise and concise language,
so we can describe a lot using fewer
words.
2. Math contains many tools and theorems
that help make general statements.
3. Math forces us to state all assumptions
explicitly and helps prevent us from
failing to acknowledge implicit
assumptions.
4. Multi-dimensionality is easily described.
Mathematical Economics vs Econometrics
Mathematical Economics
● application of mathematics to the purely
theoretical aspects of economic analysis
Econometrics
● deals with the study of empirical
observations using statistical methods of
estimation and hypothesis testing
Empirical studies and theoretical analyses are
often complementary and mutually
reinforcing:
● Theories must be tested against
empirical data for validity before they
can be applied with confidence.
●
Statistical work needs economic theory
as a guide, in order to determine the
most relevant and fruitful direction of
research.
Economic models
● A mathematical model is an abstract
description of a concrete system using
mathematical concepts and language.
● The real economy is too complex that it
is impossible for us to understand all the
interrelationships at once, hence the
need for deliberately simplified
analytical frameworks, called economic
models, that focus on the primary
factors and relationships.
“All models are wrong, but some are useful.” George Box, statistician
Ingredients of a model
● Variables
- exogenous: determined outside
the model and imposed on the
model
- endogenous: determined by the
model
Examples:
Price P, profit π, Revenue R,
cost C, national income Y, etc
● Constants
○ coefficient
Example: 7P
○ parameter
Example: aP
● Equations
○ A definitional equation set up
an identity between two
alternate expressions that have
exactly the same meaning. For
○
○
example, total profit is defined
as the excess of total revenue
over total cost; we write, π= R –
C
A behavioral equation
specifies the manner in which a
variable behaves in response to
changes in other variables. For
example, the two cost functions
■ C = 75 + 10Q
■ C = 110 + Q2 give two
different behavior for
the cost
A conditional equation states a
requirement to be satisfied. For
example, we have the
equilibrium conditions (to attain
equilibrium)
■ Qd = Qs [quantity
demanded = quantity
supplied]
■ S = I [intended saving =
intended investment]
The Real Number System
Sets
●
●
●
●
A set is a well-defined collection of
distinct objects.
The objects in a set are called the
elements or members of the set.
Capital letters A, B, C, … usually
denote sets.
Lowercase letters a, b, c, … denote the
elements of a set.
Specifying a Set
● List the elements explicitly, e.g., 𝐴 = {𝑖,
𝑜, 𝑢}
● List the elements implicitly, e.g., B ={3,
6, 9, 12, …, 90}
● Use set builder notation, e.g., C = { 𝑥 | 𝑥
= 𝑝/𝑞 where 𝑝 and 𝑞 are integers and 𝑞 ≠
0}
The Membership Relation
● Let A be a set and let x be some object.
● Notation: x ∈ A
● Meaning: x is a member of A, or x is an
element of A, or x belongs to A.
● Negated by writing x ∉ A
● Example:
V = {a, e, i, o, u}
e∈V b∉V
Equality of Sets
● Two sets A and B are equal, denoted
A=B, if they have the same elements.
● Otherwise, A≠B.
● Example: The set A of odd positive
integers is not equal to the set B of
prime numbers.
● Example: The set of odd integers
between 4 and 8 is equal to the set of
prime numbers between 4 and 8.
The Empty Set
● The set with no elements.
● Also called the null set.
● Denoted by the symbol ∅ or { }.
● Example: The set of odd numbers that
are divisible by 2.
The Universal Set
● A set U that includes all of the elements
under consideration in a particular
discussion.
● Depends on the context.
●
Examples: The set of all UP students,
the set of natural numbers, the set of
GDP factors.
Subsets
● A is a subset of B if every element of A
is an element of B.
○ Notation: A ⊆ B
● A is a proper subset of B if 𝐴 ⊆ 𝐵 and 𝐴
≠𝐵
○ Notation: A ⊂ B
Properties of Subsets
1. The null set is a subset of all sets.
2. A set is a subset of itself.
3. If A is a subset of B and B is a subset of
C, then A is a subset of C.
4. In general, if a set has n elements, a total
of 2𝑛 subsets can be formed from those
elements.
SET OPERATIONS AND VENN
DIAGRAMS
Venn Diagrams
Unions
● The union of two sets A and B is
A ∪ B = {x | x ∈ A or x ∈ B}
● The word “or” is inclusive.
Set A represented as a disk inside a rectangular
region representing U.
Possible Venn Diagrams for Two Sets
Intersections
● The intersection of A and B is
A ∩ B = {x | x ∈ A and x ∈ B}
● Example: Let A be the set of even
positive integers and B the set of prime
positive integers. Then
A ∩ B = {2}
● Definition: A and B are disjoint if
A∩B=∅
A ∩ Ac = ∅
A ∪ Ac = U
The shaded part is AC or A’
Some Laws on Set Operations
When the intersection is empty, we say Aand B
are disjoint.
Complements
If A is a subset of the universal set U, then the
complement of A is the set
Ac = { x ∈ U | x ∉ A }
CHAPTER 2 Review of Functions
Functions
Define and identify relations and functions.
Relation
A relation is any set of ordered pairs.
Function
A function is a relation in which, for each value
of the first component of the ordered pairs, there
is exactly one value of the second component.
Determining Whether Relations Are
Functions
Relations L and M are functions, because for
each different x-value there is exactly one
y-value.
In relation N, the first and third ordered pairs
have the same x-value paired with two different
y-values (6 is paired with both 2 and 5), so N is
a relation but not a function.
Using an Equation to Define a Function
In a function, no two ordered pairs can have
the same first component and different second
components.
Vertical Line Test
If every vertical line intersects the graph of a
relation in no more than one point, then the
relation represents a function.
In a function, the set of all permissible values of
the independent variable x is the domain.
The y value into which an x value is mapped is
called the image of that x value. The set of all
images is called the range of the function, which
is the set of all values that the dependent
variable y can take.
Example 2
The total cost C of a firm per day is a function of
its daily output Q: C = 150 +7Q. The firm has a
capacity limit of 100 units of output per day.
What are the domain and the range of the cost
function?
Domain = {Q | 0 ≤ Q ≤ 100}
The minimum C value at 150 (when Q = 0) and
the maximum C value at 850 (when Q = 100),
and we have the Range = {C | 150 ≤ C ≤ 850}
Beware, however, that the extreme values of the
range may not always occur where the extreme
values of the domain are attained.
FUNCTION OF TWO VARIABLES
● A function f of two variables is a rule
that assigns to each ordered pair of real
numbers (x, y) in a set D a unique real
number denoted by f (x, y).
● The set D is the domain of f.
● Its range is the set of values that f takes
on, that is,
{f(x, y) | (x, y) ∈ D}
● We often write z = f(x, y) to make
explicit the value taken on by f at the
general point (x, y). The variables x and
y are independent variables. z is the
dependent variable.
● A function of two variables is just a
function whose:
○ Domain is a subset of R2
○ Range is a subset of R
● One way of visualizing such a function
is by means of an arrow diagram, where
the domain D is represented as a subset
of the xy-plane.
Example:
● Suppose that output is determined by the
amounts of capital (K) and labor (L)
employed; then we can write a
production function in the general form:
● Q = Q(K, L)
Extension to higher dimensions
● F = F(x,y,z)
● G = G(w,x,y,z)
● And so on……
CHAPTER 3 Equilibrium
●
“Equilibrium is a constellation of selected
interrelated variables so adjusted to one another
that no inherent tendency to change prevails in
the model which they constitute.”
●
More generally, it means that from the various
feasible and available choices, choose “best” one
according to a certain criterion, and the one
being finally chosen is called an equilibrium.
The study of what the equilibrium state is like is
referred to as statics.
●
●
●
Note: equilibrium is a positive (as
opposed to normative) economic
concept. There is nothing inherently
good or bad about equilibrium.
Even though a certain equilibrium
position may represent a desirable state
and something to be striven for—such
as a profit-maximizing situation, from
the firm’s point of view—another
equilibrium position may be quite
undesirable and therefore something to
be avoided, such as an
underemployment equilibrium level of
national income.
The only warranted interpretation is that
an equilibrium is a situation which, if
attained, would tend to perpetuate itself,
barring any changes in the external
forces.
𝑄𝑠 = the quantity supplied of the
commodity;
𝑃 = the price of the commodity
The two equations:
● 𝑄𝑑 = 𝑎 − 𝑏𝑃 (𝑎, 𝑏 > 0)
● 𝑄𝑠 = −𝑐 + 𝑑𝑃 (𝑐, 𝑑 > 0)
The equilibrium condition:
𝑄𝑑 = 𝑄𝑠
The Model
● 𝑄𝑑 = 𝑎 − 𝑏𝑃 (𝑎, 𝑏 > 0)
● 𝑄𝑠 = −𝑐 + 𝑑𝑃 𝑐, 𝑑 > 0
● 𝑄𝑑 = 𝑄𝑠
Goal:
● We want to obtain the specific values 𝑄∗
and 𝑃∗ satisfying the system of
equations.
● Note that 𝑄∗ is the common specific
value for 𝑄𝑑 and 𝑄𝑠, ie,
𝑄∗ = 𝑄𝑑∗ = 𝑄𝑠∗
Graphical Solution
Partial Market Equilibrium- A Linear Model
Partial-equilibrium market model is a model of
price determination in an isolated market for a
commodity.
The three variables:
● 𝑄𝑑 = the quantity demanded of the
commodity;
Algebraic Solution
From 𝑄𝑑 = 𝑄𝑠, we have
𝑎 − 𝑏𝑃 = −𝑐 + 𝑑𝑃
Thus 𝑏 + 𝑑 𝑃 = 𝑎 + 𝑐. Since 𝑏 + 𝑑 ≠ 0, then the
equilibrium price is
𝑷∗ = 𝒂 + 𝒄 / 𝒃 + 𝒅
The equilibrium quantity can be obtained by
substituting 𝑃∗ into either 𝑄𝑑 or 𝑄𝑠:
𝑸∗ = 𝒂𝒅 − 𝒃𝒄 / 𝒃 + 𝒅
Note that to be economically meaningful, the
model should contain the additional restriction
that 𝑎𝑑 > 𝑏𝑐.
Partial Market Equilibrium- A NonLinear
Model
The partial market model can be nonlinear
Suppose a model is given by
● 𝑄𝑑 = 𝑄𝑠
● 𝑄𝑑 = 4 − 𝑃2
● 𝑄𝑠 = 4𝑃 − 1
Algebraic Solution
Equating, we have 4 − 𝑃2 = 4𝑃 − 14
Which reduces to the quadratic equation 𝑃2 + 4𝑃
− 5 = 0.
We then use the quadratic formula to obtain 𝑷∗ =
𝟏 or 𝑷∗ = −𝟓 (not admissible).
Correspondingly, we have 𝑸∗ = 𝟑.
Graphical Solution
General Market Equilibrium
Rationale
● We have discussed methods of an
isolated market, wherein the 𝑄𝑑 and 𝑄𝑠
of a commodity are functions of the
price of that commodity alone.
● A more realistic model for the demand
and supply function of a commodity
should take into account the effects not
only of the price of the commodity itself
but also of the prices of other
commodities
Equilibrium Condition
● The equilibrium condition of an
n−commodity market model will
involve n equations, one for each
commodity, in the form:
𝐸𝑖 = 𝑄𝑑𝑖 − 𝑄𝑠𝑖 = 0 (i = 1, 2, · · · , n)
where 𝑄𝑑𝑖= 𝑄𝑑𝑖(𝑃1,𝑃2 … , 𝑃𝑛) and
𝑄𝑠𝑖= 𝑄𝑠𝑖(𝑃1,𝑃2 … ,𝑃𝑛)
are the demand and supply functions of
commodity i, and 𝑃1, 𝑃2 … ,𝑃𝑛 are
prices of commodities
Thus solving n equations 𝐸𝑖 = 𝑄𝑑𝑖 − 𝑄𝑠𝑖 = 0
where (i = 1, 2, · · · , n), we obtain the n
equilibrium prices 𝑷𝒊∗ if a solution does indeed
exist.
And then the 𝑸𝒊∗ may be derived from the
demand or supply functions
Two-Commodity Market Model
The Model
● 𝑄𝑑1 − 𝑄𝑠1 = 0
● 𝑄𝑑1 = 𝑎0 + 𝑎1𝑃1 + 𝑎2𝑃2
● 𝑄𝑠1 = 𝑏0 + 𝑏1𝑃1 + 𝑏2𝑃2
● 𝑄𝑑2 − 𝑄𝑠2 = 0
● 𝑄𝑑2 = α0 + α1𝑃1 + α2𝑃2
● 𝑄𝑠2 = β0 + β1𝑃1 + β2𝑃2
where the a and b coefficients pertain to the
demand and supply functions of the first
commodity, and the α and β coefficients are
assigned to those of the second.
●
By substituting the second and third
equations into the first (for the first
commodity) ; and the fifth and sixth
equations into the fourth (for the second
commodity), the model is reduced to
two equations in two variables:
(𝑎0−𝑏0) + (𝑎1 − 𝑏1)𝑃1 + (𝑎2−𝑏2)𝑃2=0
(α0−β0) + (α1 − β1)𝑃1 + (α2−β2)𝑃2=0
To simplify, we let
𝑐𝑖 = 𝑎𝑖 − 𝑏𝑖 (i = 0, 1, 2)
γ𝑖 = α𝑖 − β𝑖 (i = 0, 1, 2)
and we have,
𝑐1𝑃1 + 𝑐2𝑃2 = −𝑐0
γ1𝑃1 + γ2𝑃2 = −γ0
Solving, we obtain,
For it to make sense, we add restrictions:
First, 𝑐1γ2 − 𝑐2γ1 ≠ 0.
Second, to assure positivity, the numerator must
have the same sign as the denominator. From
these, we can obtain the values for 𝑄1∗ and 𝑄2∗
Problem Suppose that the demand and supply
functions are numerically as follows:
• 𝑄𝑑1 = 10 − 2𝑃1 + 𝑃2
• 𝑄𝑠1 = −2 + 3𝑃1
• 𝑄𝑑2 = 15 + 𝑃1 − 𝑃2
• 𝑄𝑠2 = −1 + 2𝑃2
What are the equilibrium prices and quantities?
Answers
• 𝑃1 ∗ = 52/14
• 𝑃2 ∗ = 92/14
• 𝑄1 ∗ = 64/7
• 𝑄2 ∗ = 85/7.
CHAPTER 4 Linear Models and Matrix
Algebra
Matrix algebra can enable us to do many things:
(1) It provides a compact way of writing an
equation system, even an extremely
large one.
(2) It leads to a way of testing the existence
of a solution by evaluation of a
determinant – a concept closely related
to that of a matrix.
(3) It gives a method of finding that solution
if it exists.
Matrix and Vectors
In general, a system of m linear equations in n
variables (x1, x2, · · · , xn) can be arranged into
such formula
where the double-subscripted symbol aij
represents the coefficient appearing in the ith
equation and attached to the jth variable xj and dj
represents the constant term in the jth equation.
Example: The two-commodity linear market
model can be written – after eliminating the
quantity variables – as a system of two linear
equations.
Matrix as Arrays
Three types of ingredients in the equation
system
1. set of coefficients aij
2. set of variables x1, x2, · · · , xn
3. set of constant terms d1, d2, · · · , dm