Econ 413: Advanced Economic Analysis

Current Semester: Spring 2010

Section 01:
Meets T R from 3:45pm - 5:00pm in HH 1300

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

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

1: Introduction

Introduction: Notation and basic set theory; functions.

2: Linear Algebra

Topics in Linear Algebra include:
Systems of equations and matrices; Gauss-Jordan Elimination; Matrix Algebra; Formal definition of a matrix. Symmetric Matrices. Row space, column space and null space. Rank and nullity. Fundamental Theorem of Linear Algebra. Determinants. Cramer’s Rule; Definiteness of a matrix.

3. Euclidean Spaces

Topics in Euclidean Spaces include:
Vectors. Inner Product. Distance between vectors; Vector Spaces (Euclidean Spaces) and Subspaces; Hyperplanes. Budget Sets and Simplexes; Linear Combinations and Spanning Sets; Linearly Independent Sets. Basis and Dimension. Eigenvalues and Eigenvectors.

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.

Problem Sets

Problem Sets will be posted here as we proceed through the semester

Exams

Exams and their solutions will be posted here.