Undergraduate Mathematical Economics Lecture 1

Undergraduate MathematicalEconomicsLecture 1

Yu Ren

WISE, Xiamen University

September 15, 2014

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Courses Description and RequirementLinear Algebra

Outline

1 Courses Description and Requirement

2 Linear Algebra

Yu Ren Undergraduate Mathematical Economics Lecture 1

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Course Outline

mathematical techniques used ineconomics coursesMathematics for Economists by Simon andBlume


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Course Outline

mathematical techniques used ineconomics coursesMathematics for Economists by Simon andBlume


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Course Outline

15 weeks, including one midterm examgrading policy

quiz: 30 %midterm exam:30%final exam:40%


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Course Outline

15 weeks, including one midterm examgrading policy

quiz: 30 %midterm exam:30%final exam:40%


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Quiz

7 quizzes in totaleach has two questions, covering thematerials of the previous lecturerequired to finish it within 20 minutes


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Quiz



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Quiz



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Office hours

my office hours: 2:30-4:00pm TuesdayTA office hours: TBAany courses-related questions can beasked


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Some Questions

1 Why economics need mathematics?2 What is the focus of this course?3 How to achieve good performances in this

course


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Some Questions


course


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Some Questions


course


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Linear Algebra

Linear AlgebraChapter 6, 7, 8, 9


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Linear Equation

an equation is linear if

a1x1 + a2x2 + · · ·+ anxn = b

ai : parameters, xi : variables (i = 1,2, · · · ,n)


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Linear Equation

For examplex1 + 2x2 = 32x1 − 3x2 = 8


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Linear Systems

a11x1 + a12x2 + · · · + a1nxn = b1a21x1 + a22x2 + · · · + a2nxn = b2

... +... + · · · +

... =...

am1x1 + am2x2 + · · · + amnxn = bm


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Example: Tax Benefits

Before tax profits: 100,00010% of after-tax profits to Red Crossstate tax of 5% of its profits after Red Crossdonationfederal tax of 40% of its profits afterdonation and state tax


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Example: Tax Benefits

Question: How much does the companypay in state taxes, federal taxes and RedCross?


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Example: Answer

C: (Red Cross); S: (State taxes); F: (Federaltaxes)

C = (100,000− S − F ) ∗ 0.1 (1)S = (100,000− C) ∗ 0.05 (2)

F = (100,000− C − S) ∗ 0.4 (3)


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General Questions

Questions:1 Does a solution exist?2 How many solutions are there?3 Is there an efficient algorithm that computes

actual solutions?


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General Questions

method to find the answers1 substitutions2 elimination of variables3 matrix methods


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Substitution

Taught in high school classSteps:

1 m equations, n variables2 solve xn in terms of the other variables3 substitute this expression for xn into the other

equations4 a new system of m − 1 equations and n − 1

variables5 repeat the process


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Substitution






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Substitution






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Substitution






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Substitution






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Elimination


1 m equations, n variables2 multiplying both sides of an equation by a

nonzero number3 adding one equation to another equation in

order to eliminate one variable


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Elimination






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Elimination






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Elimination

Example (Page 126):

x1 − 0.4x2 − 0.3x3 = 130−0.2x1 + 0.88x2 − 0.14x3 = 74−0.5x1 − 0.2x2 + 0.95x3 = 95


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Matrix Methods: Elementary RowOperations

augmented matrix: add on a columncorresponding to the right-hand side insystem

A =

a11 a12 · · · a1n b1a21 a22 · · · a2n b2... ... · · · ... ...

am1 am2 · · · amn bm


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Matrix Methods: Elementary RowOperations

interchange two rows of a matrixchange a row by adding to it a multiple ofanother rowmultiply each element in a row by the samenonzero number


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Definitions

leading zeros: a row of a matrix is said tohave k leading zeros if the first k elementsof the row are all zeros and the k + 1element of the row is not zero


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Definitions

row echelon form: a matrix is in rowechelon form if each row has more leadingzeros than the row preceding it. Example: 1 2 3

0 2 30 0 3

row echelon form can be obtained byelementary row operations


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Rank

Rank: the rank of a matrix is the number ofnonzero rows in its row echelon form.a matrix→ row echelon form→ rank


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Rank

Let A be the coefficient matrix of somelinear systems and let A be thecorresponding augmented matrix. Then thissystems has a solution if and only ifrank(A)=rank(A).


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Matrix Algebra: Addition

a11 · · · a1n... aij

...ak1 · · · akn

+

b11 · · · b1n... bij

...bk1 · · · bkn

=

a11 + b11 · · · a1n + b1n... aij + bij

...ak1 + bk1 · · · akn + bkn


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Matrix Algebra: Subtraction

a11 · · · a1n... aij

...ak1 · · · akn

− b11 · · · b1n

... bij...

bk1 · · · bkn

=

a11 − b11 · · · a1n − b1n... aij − bij

...ak1 − bk1 · · · akn − bkn


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Matrix Algebra: scalar multiplication

r

a11 · · · a1n... aij

...ak1 · · · akn

=

ra11 · · · ra1n... raij

...rak1 · · · rakn


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Matrix Algebra: matrix multiplication

We can define the matrix product AB if and onlyif number of column of A = number of rowsof B .(i , j) entry of AB is

(ai1 ai2 · · · aim

)b1jb2j...

bmj

=m∑

h=1

aihbhj


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Laws of Matrix Algebra

Associative Laws:(A+B) +C = A+ (B +C); (AB)C = A(BC)

Commutative Law for addition:A + B = B + ADistributive Laws: A(B + C) = AB + AC;(A + B)C = AC + BC


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Transpose

transpose:a11 a12 · · · a1na21 a22 · · · a2n... ... aij

...ak1 ak2 · · · akn

T

=

a11 a21 · · · ak1a12 a22 · · · ak2... ... aji

...a1n a2n · · · akn


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Transpose

(A + B)T = AT + BT ; (A− B)T = AT − BT ;(AT )T = A; (rA)T = rAT ; (AB)T = BT AT


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Special Matrices

Page 160Square Matrix; Column Matrix; Row Matrix;Diagonal Matrix; Upper-Triangular Matrix;Lower-Triangular Matrix; Symmetric Matrix;Permutation MatrixIdempotent Matrix; Nonsingular Matrix


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Inverse

Let A be a n × n matrix. Matrix B is aninverse of A if AB = BA = I. If the matrix Bexists, we say A is invertible.


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Inverse

Theorem 8.5: An n × n matrix can have atmost one inverse.(A−1)−1 = A; (AT )−1 = (A−1)T ; AB isinvertible and (AB)−1 = B−1A−1

How to find the inverse matrix? Hold thisquestion for a while.


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Inverse




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Inverse




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Determinants

det(a) = a

det(

a11 a12a21 a22

)= a11a22 − a12a21


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Determinants

det(a) = a

det(

a11 a12a21 a22

)= a11a22 − a12a21


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Minor and Determinant

Let A be an n × n matrix. Let Aij be the(n − 1)× (n − 1) submatrix obtained bydeleting row i and column j from A. Thenumber Mij = det(Aij) is called the (i , j)minor of A and the scalar Cij = (−1)i+jMij iscalled the (i , j) cofactor of A.


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Determinant of an n× n matrix A is given by

det(A) = a11C11 + a12C12 + · · ·+ a1nC1n

Example 9.2 (Page 192)


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Determinant of an n× n matrix A is given by

det(A) = a11C11 + a12C12 + · · ·+ a1nC1n

Example 9.2 (Page 192)


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Uses of the Determinant

For any n × n matrix A, let Cij denotes the(i , j)th cofactor of A. The n × n matrixwhose (j , i) entry is Cij is called the adjointof A, denoted by adj(A).


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A−1 = 1det(A)adj(A)


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(Cramer’s Rule) the unique solutionx = (x1, · · · , xn) of the n × n system Ax = bis xi =

det(Bi)det(A) , where Bi is the matrix A with

the right-hand side b replacing the i thcolumn of AExample 9.3 (Page 195)


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(Cramer’s Rule) the unique solutionx = (x1, · · · , xn) of the n × n system Ax = bis xi =

det(Bi)det(A) , where Bi is the matrix A with

the right-hand side b replacing the i thcolumn of AExample 9.3 (Page 195)


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Some Properties of Determinant

det(AT ) = det(A)det(AB) = (det(A))(det(B))

det(A + B) 6= det(A) + det(B)


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Undergraduate Mathematical Economics Lecture 1