SUMMARY OF PROOFS OF OPENNESS, CLOSEDNESS AND COMPACTNESS
MING LI
Consider the metric space (X, d). Definition 1: A set A X is said to be open, if for any x A, there exists r > 0, such that the open ball B(x, r) A. Definition 2-1: A set A X is said to be

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 21
Fall 2009
Higher Order Linear Difference and Differential Equations We will consider second-order difference and differential equations. The dynamics of solutions depend on the sol

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 22
Fall 2009
The System of First-Order Difference and Differential Equations When we want to introduce interactions among variables, we will consider the system of difference and diff

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 3 Open and Closed Sets
Fall 2009
In optimization problems we often assume that constraint sets are compact for a reason that we will explain later. In the Euclidean space a set is com

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 4
Fall 2009
Continuous Functions and Existence of Optimal Solutions We will present sufficient conditions for an optimization problem to have a solution. Outline A. Continuous Functio

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 5 Local and Global Optima
Fall 2009
We will review basic concepts in optimization and differential calculus for univariate functions. Our goal is to characterize optimal solutions usi

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 6 Vectors and Matrices
Fall 2009
To characterrize optimal solutions for multivariate objective functions, it is convenient to use vectors and matrices. Outline A. Vectors B. Matrices

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 7
Fall 2009
First and Second Order Conditions for Multivariate Objective Functions We now characterize first and second order conditions by using gradient vectors and Hessian matrices

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 8 Determinantal Tests of Definiteness
Fall 2009
To test the second-order sufficient condition we need to find out the definiteness of the Hessian matrix of an objective function. Ther

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 9 Eigenvalue Tests of Definiteness
Fall 2009
We will review another test for definiteness to check the second-order sufficient condition for optimization. As shown later in the semest

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Problem Set 3 Due in class on Tuesday, September 15
Fall 2008
Do exercise 12.21 in Simon and Blume (1994, p.269270). For each of the five sets in 2 , provide a formal proof to justify your answer.

Atsushi Inoue
ECG 765: Mathematics for Economists Problem Set 4 Due in class on Tuesday, September 22
Fall 2009
1. A fishery earns a profit of (x) from catching and selling x units of fish. The firm owns a pool which currently has y1 fish in it. If x [0,

Atsushi Inoue
ECG 765: Mathematics for Economists Problem Set 5 Due in class on Tuesday, October 6 in class
Fall 2009
1. Consider a lexicographic utility function U : two properties:
that satisfies
(i) For any (x, y) and (x , y ) such that x > x , U (x, y

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Problem Set 6 Due in class on Tuesday, October 13
Fall 2009
1. Let
A=
a11 a21 . . .
a12 a22
am1 am2
a1n a2n . . . . . . amn
,
x=
b1 b2 . . . bn
.
Write Ax using aij , bj and the notation.

Atsushi Inoue
ECG 765: Mathematics for Economists Problem Set 7 Due in class on Tuesday, October 20
Fall 2009
1. Examine the definiteness or semidefiniteness of the following matrix: 1 2 3 A = 2 4 6 . 3 6 0 (2 points) 2. Suppose that matrix A has an eigen

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 20 First-Order Differential Equations
Fall 2009
Differential equations are used to model in continuous time. We will consider first-order differential equations and discuss how to obt

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 2 Sequences and Limits
Fall 2009
To characterize objective functions and constraint sets in optimization problems, we need to introduce some mathematical tools. Outline A. Functions B

Math 233
Hessians andUnconstrainedOptimization
Fall 2001
The Big Picture: Secondderivatives,whetherin single or multi variablecalculus,measuretherateof changein slopes(i.e.thecurvatureof thefunctionf). What makesproblemsharderin multivariablecalcis that w

CSC228 Sequences/ Summations 1. What is the term a8 of the sequence cfw_ a n if a n equals a. 2 n + 1 n b. 2
5/1/2009
2. (Rosen, page 236) List the first 10 terms of the sequence that begins with 3 with each succeeding term is twice the previous term. 3.

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Midterm Exam
Spring 1999
I. Let x = (x1 , x2 , ., xn ) be an n-dimensional column vector and A = cfw_aij be an n n symmetric matrix. Define f (x) = -x Ax and g(x) = x x. Both f and g are defined o

Atsushi Inoue ECG 765: Mathematical Methods for Economics Sp Midterm Exam 2 1. (a) Compute the derivative of f (x) = ln(x2 + 4x + 3) (5 points). (b) Find the first three terms in the Taylor series approximation of x f (x) = 1+x around the point x = 0 (5 p

Atsushi Inoue
Econ 765: Mathematical Methods for Economists Problem Set 2
Fall 2009
Due in class on Tuesday, September 8, at 11:45am When you answer each of the following questions, please use formal arguments (e.g., use definitions and theorems rather th

Atsushi Inoue
Econ 765: Mathematical Methods for Economists Problem Set 1
Fall 2009
Due in class on Tuesday, September 1, at 11:45am 1. A propositional form is called a tautology if it is always true. A proposition form is said to be a contradiction if it

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 11 Concave and Convex Functions
Fall 2009
So far we have focused on finding local optima using first-order and secondorder conditions. Local optima are not necessarily global optima i

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 12
Fall 2009
Matrix Inverse, Matrix Rank and the Fundamental Theorem of Linear Algebra When we solve constraint optimization problems, one of the conditions, called constraint qualifi

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 13 Lagrange's Method
Fall 2009
Many problems in economics can be written as constrained optimization problems. First we will consider equality constraint optimization problems. Outlin

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 14
Fall 2009
Second-Order Conditions for Equality-Constrained Optimization Because the Lagrange theorem provides first-order necessary conditions for constrained local optima only, th

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 15 Inequality-Constrained Optimization
Fall 2009
When constraints involve inequalities, the Kuhn-Tucker theorem provides first-order necessary conditions for constraint local optima.

Atsushi Inoue
ECG 765: Mathematical Methods for Economics Lecture Notes 16
Fall 2008
Optimization Subject to Mixed Constraints We will consider cases in which there are equality constraints as well as inequality constraints in maximization and minimizatio