Econ 413: Advanced Economic Analysis

Current Semester: Spring 2010

Section 01:
Meets T R from 3:45pm - 5:00pm in HH 1301

This course is an advanced course in mathematical economics that is intended to provide a preview of the technical rigors involved in graduate studies. As such, this course will expose the student to the use of applied mathematics in solving economic problems. We will explore advanced techniques used to study both microeconomic and macroeconomic problems, whilst consolidating the mathematical techniques needed for graduate work. Thus, students intending to go on to graduate school (in economics), or any student who is interested in brushing up on their math skills or interested in exploring the rigors of technical analysis will find this course extremely beneficial.

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Syllabus

Lectures

Required Text: Fundamental Methods of Mathematical Economics, by Alpha Chiang and Kevin Wainwright, 4th Edition, 2005, McGraw Hill. ISBN: 0-07-010910-9.

Alternative Text: Mathematical Methods for Economics, by Michael Klein, 2nd Edition, 2002, Addison Wesley/Prentice Hall. ISBN: 0201726262

In order to access the lecture notes below, you will need your net-id and password.

This is a brief outline of the topics to be covered in lectures:

Topic	Lecture Notes		Readings: Chiang and Wainwright (CW); Klein (K)
	(Singles)	(4 to a page)
Introduction: Notation and basic set theory; Introduction to functions.	Lecture 1	Lecture 1	CW: Chapter 1 - 2, 10 K: Chapter 1 - 3
Linear Algebra	Lecture 2a Lecture 2b *updated* Lecture 2c	Lecture 2a Lecture 2b *updated* Lecture 2c	CW: Chapters 4 - 5 K: Chapters 4 - 5
Euclidean Spaces	Lecture 3	Lecture 3	CW: Chapter 4, 11.3 K: Chapter 4
Calculus	Lecture 4	Lecture 4	CW: Chapters 6 - 8 K: Chapters 6 - 8
Optimization	Lecture 5		CW: Chapters 9, 11-13 K: Chapters 9 - 11
Difference and Differential Equations	Lecture 6		CW: Chapters 14 - 18 K: Chapters 12 - 14
Dynamic Optimization	Lecture 7		CW: Chapter 20 K: Chapter 15
Introduction to Real Analysis	Lecture 8		Rudin

Detailed Description of Topics

1: Introduction

Introduction to mathematical economics; Notation; Numbers; Basic Set Theory; Introduction to functions

2: Linear Algebra

Topics in Linear Algebra include:

Part 1:
Introduction to matrices, Matrix operations, Solving systems of equations using matrices: Gaussian Elimination; Matrix Algebra; Properties of matrices; Symmetric Matrices.

Part 2:

Vectors; Vector Spaces (Euclidean Spaces) and Subspaces; Linear Combinations and Spanning Sets; Linearly Independent Sets; Basis and Dimension; Row space, column space and null space. Rank and nullity. Fundamental Theorem of Linear Algebra.

Part 3:

Determinants. Calculating Inverse Matrices; Cramer’s Rule.

Part 4: Application of Linear Algebra to Economics - Econometrics.

3. Euclidean Spaces

Topics in Euclidean Spaces include:
Vectors. Inner Product. Distance between vectors; Hyperplanes. Budget Sets and Simplexes; Eigenvalues and Eigenvectors. Quadratic Forms: Definiteness of a matrix.

4. Multivariate Calculus

Topics in Multivariate Calculus include:
Differentiation at a point. Partial Differentiation. Gradients and directional derivatives. Derivative matrix (Jacobian). Differentiation and continuity. The Chain Rule. Higher order derivatives. Young’s Theorem. Hessian Matrix. Implicit Differentiation and the Implicit Function Theorem.

5. Optimization

Topics in optimization include:
Quadratic Forms; Definiteness of Quadratic Forms; Local vs. absolute maximum; Unconstrained Maximum; Constrained local maximum. Kuhn-Tucker optimization. Convexity and concavity. Concavity and optimization. Quasiconcavity and quasiconvexity. The Envelope Theorem. The Envelope Theorem with constrained optimization.

6. Difference and Differential Equations

Topics include:
Introduction to differential equations and boundary value problems. Higher-order differential equations. Lag and difference operators. Linear first-order difference equations. Boundary conditions. ARMA representations.

7. Dynamic Optimization

Topics include:
Calculus of Variations; Euler equations; Boundary Conditions; Transversality condition. Introduction to Optimal Control Theory; The Maximum Principle; State variables, Controls, and Laws of Motion; Hamiltonian Functions.

8: Real Analysis (Time Permitting)

Topics in Real Analysis include: Ordered Sets. Upper and Lower Bounds. Supremeum and infimium. Metrics and distance functions. Neighborhoods, interior and limit points. Open and Closed Sets. Bounded Sets. Interior and Closure of a set. Compact and Connected Sets. Sequences, subsequences. Cauchy Sequences. Series, geometric series.

Assignments

A list of assignments will be posted here as we proceed through the semester. You will need your net-id and password to access them.

Assignment 1: Practice questions on sets and functions. Solutions

Assignment 2: Matrices and Matrix Operations 1 Solutions

Assignment 3: Single and Multivariate Calculus; Comparative Statics

Exams

Exams and their solutions will be posted here.

Quizzes

Quiz 1 Quiz 1 Solutions

Quiz 2 Quiz 2 Solutions

Midterm

Midterm Exam Due March 16th 2010

Midterm Exam Solutions