Graduate Mathematical Economics Lecture 1

Graduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 23, 2012

Outline 1 2

Course Outline ematical techniques used in graduate level economics courses Mathematics for Economists by Simon and Blume 15 weeks, including one midterm exam (week 8) grading policy quiz: 30 % midterm exam:30% final exam:40%

Quiz one quiz every two weeks two questions within 20 minutes based on the exercises in the textbook

Office hours my office hours: Monday 2:30-4:00pm TA office hours: TBA any courses-related questions can be asked try to make full use of them

Some Questions 1 Why economics need ematics? 2 What is the focus of this course?

Chapter 6, 7, 8, 9

Linear Equation an equation is linear if a 1 x 1 + a 2 x 2 + + a n x n = b a i : parameters, x i : variables (i = 1, 2,, n)

Linear Equation For example x 1 + 2x 2 = 3 2x 1 3x 2 = 8

Linear Systems a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. +. + +. =. a m1 x 1 + a m2 x 2 + + a mn x n = b m

Example: Tax Benefits Before tax profits: 100,000 10% of after-tax profits to Red Cross state tax of 5% of its profits after Red Cross donation federal tax of 40% of its profits after donation and state tax

Example: Tax Benefits Question: How much does the company pay in state taxes, federal taxes and Red Cross?

Example: Answer C: (Red Cross); S: (State taxes); F: (Federal taxes) C = (100, 000 S F ) 0.1 (1) S = (100, 000 C) 0.05 (2) F = (100, 000 C S) 0.4 (3)

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

General Questions method to find the answers 1 substitutions 2 elimination of variables 3 matrix methods

Substitution Taught in high school class Steps: 1 m equations, n variables 2 solve x n in terms of the other variables 3 substitute this expression for x n into the other equations 4 a new system of m 1 equations and n 1 variables 5 repeat the process

Elimination Taught in high school class Steps: 1 m equations, n variables 2 multiplying both sides of an equation by a nonzero number 3 adding one equation to another equation in order to eliminate one variable

Elimination Example (Page 126): x 1 0.4x 2 0.3x 3 = 130 0.2x 1 + 0.88x 2 0.14x 3 = 74 0.5x 1 0.2x 2 + 0.95x 3 = 95

Matrix Methods: Elementary Row Operations augmented matrix: add on a column corresponding to the right-hand side in system Â = a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2.... a m1 a m2 a mn b m

Matrix Methods: Elementary Row Operations interchange two rows of a matrix change a row by adding to it a multiple of another row multiply each element in a row by the same nonzero number

Definitions leading zeros: a row of a matrix is said to have k leading zeros if the first k elements of the row are all zeros and the k + 1 element of the row is not zero

Definitions row echelon form: a matrix is in row echelon form if each row has more leading zeros than the row preceding it. Example: 1 2 3 0 2 3 0 0 3 row echelon form can be obtained by elementary row operations

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

Rank Let A be the coefficient matrix of some linear systems and let Â be the corresponding augmented matrix. Then this systems has a solution if and only if rank(â)=rank(a).

Matrix Algebra: Addition a 11 a 1n. a ij. a k1 a kn = + b 11 b 1n. b ij. b k1 b kn a 11 + b 11 a 1n + b 1n. a ij + b ij. a k1 + b k1 a kn + b kn

Matrix Algebra: Subtraction a 11 a 1n. a ij. a k1 a kn = b 11 b 1n. b ij. b k1 b kn a 11 b 11 a 1n b 1n. a ij b ij. a k1 b k1 a kn b kn

Matrix Algebra: scalar multiplication r a 11 a 1n. a ij. a k1 a kn = ra 11 ra 1n. ra ij. ra k1 ra kn

Matrix Algebra: matrix multiplication We can define the matrix product AB if and only if number of column of A = number of rows of B. (i, j) entry of AB is ( ) ai1 a i2 a in b 1j b 2j. b mj m = h=1 a ih b hj

Laws of Matrix Algebra Associative Laws: (A + B) + C = A + (B + C); (AB)C = A(BC) Commutative Law for addition: A + B = B + A Distributive Laws: A(B + C) = AB + AC; (A + B)C = AC + BC

Transpose transpose: a 11 a 12 a 1n a 21 a 22 a 2n.. a ij. a k1 a k2 a kn T = a 11 a 21 a k1 a 12 a 22 a k2.. a ji. a 1n a 2n a kn

Transpose (A + B) T = A T + B T ; (A B) T = A T B T ; (A T ) T = A; (ra) T = ra T ; (AB) T = B T A T

Special Matrices Page 160 Square Matrix; Column Matrix; Row Matrix; Diagonal Matrix; Upper-Triangular Matrix; Lower-Triangular Matrix; Symmetric Matrix; Permutation Matrix Idempotent Matrix; Nonsingular Matrix

Inverse Let A be a n n matrix. Matrix B is an inverse of A if AB = BA = I. If the matrix B exists, we say A is invertible.

Inverse Theorem 8.5: An n n matrix can have at most one inverse. (A 1 ) 1 = A; (A T ) 1 = (A 1 ) T ; AB is invertible and (AB) 1 = B 1 A 1 How to find the inverse matrix? Hold this question for a while.

Determinants det(a) = a ( ) a11 a det 12 = a a 21 a 11 a 22 a 12 a 21 22

Minor and Determinant Let A be an n n matrix. Let A ij be the (n 1) (n 1) submatrix obtained by deleting row i and column j from A. The number M ij = det(a ij ) is called the (i, j) minor of A and the scalar C ij = ( 1) i+j M ij is called the (i, j) cofactor of A.

Minor and Determinant Determinant of an n n matrix A is given by det(a) = a 11 C 11 + a 12 C 12 + + a 1n C 1n Example 9.2 (Page 192)

Uses of the Determinant For any n n matrix A, let C ij denotes the (i, j)th cofactor of A. The n n matrix whose (j, i) entry is C ij is called the adjoint of A, denoted by adj(a).

Uses of the Determinant A 1 = 1 det(a) adj(a)

Uses of the Determinant (Cramer s Rule) the unique solution x = (x 1,, x n ) of the n n system Ax = b is x i = det(b i) det(a), where B i is the matrix A with the right-hand side b replacing the ith column of A Example 9.3 (Page 195)

Some Properties of Determinant det(a T ) = det(a) det(ab) = (det(a))(det(b)) det(a + B) det(a) + det(b)