**Introduction to Quantitative Economics**.

ECO 6371.

**Fall, 2015.**** **

Instructor: Santanu Roy.

Office: 301-A Umphrey Lee.

Tel: 214 768 2714

Fax: 214 768 1821

E-mail: sroy@smu.edu

Office Hours: Mondays and Wednesdays, 10 - 11.30 AM & by appointment.

Lectures: Tuesdays and Thursdays, 9.30 - 10.50 AM, Room 303, Umphrey Lee.

Teaching Assistant: Mr. Xiang Han

E-mail: xiangh@smu.edu

Office Hours:: Tuesday and Thursday, 3.30 -
4.30 PM, in the department library

*General Objective:*

This is a course on basic mathematics for a doctoral program in economics.

There is high variation in the extent and kind of mathematics used in various sub-fields and approaches in economic research (including econometrics). It is impossible for a course of this kind to cover all the mathematical results used in the major areas of economics - or even most of them. A course of this sort has to be selective and limited.

This course will focus on some basic results in linear algebra, real analysis and static optimization theory. The selection of topics is, to a large extent, motivated by constraints on the time available for the course, the typical background of the incoming Ph.D. students and the need to have a course that is self-contained. Therefore, the course will not be able to cover topics such as probability/measure theory and stochastic dynamic optimization. Hopefully, other courses will cover such material.

Instead of presenting a haphazard glossary of large number of results, the course will aim to provide students with an understanding of the important results, the mathematical arguments behind them and their implications.

Learning Objectives:

* Acquaint students with a set of fundamental mathematical results and techniques that are useful for other core and field courses in the Ph.D. program in economics.

* Equip them with the ability to construct and analyze
mathematical economic models.

* Enable them to handle constrained and unconstrained optimization problems using classical and modern approaches.

* Create understanding of some basic concepts in linear
algebra that are useful in econometric analysis.

* Enable them to read scientific papers in their areas of interest.

*Reading*:* *

There is no required textbook for this course.

The course will be based on a highly acclaimed set of lecture notes by Prof. T. Mitra of Cornell University for a similar course.

Copies of these notes will be made available to students (for private use only).

Recommended supplementary reading (optional):

*·** **Mathematics**
for Economists *by Carl Simon and Lawrence Blume, W. W. Norton
& Co., 1994. ISBN 0-393-95733-0.

*·** **A**
First Course in Optimization Theory* by Rangarajan
Sundaram, Cambridge University Press, 1996. ISBN 0-521-49770-1.

*Home Assignments:*

Home assignments will be given out from time to time. It is extremely important that students work on these assignments on their own. Solutions will be circulated or discussed in class.

*Examinations*:

One midterm and one final exam.

*Evaluation: *

Final grade will be assigned on the following basis:

Midterm Exam (50%)

Final exam (50%).

__Tentative List of Topics__

I. Linear Algebra.

· Vectors:
Vector Spaces, Linear Dependence, Rank and Basis, Inner Product and Norm.

· Matrices:
Matrix Algebra, Rank, Inverse, Invertible and Non-Singular Matrix.

· Simultaneous
Linear Equations: Existence, Uniqueness and Calculation of solutions,
Determinants, Matrix Inversion, Cramer’s Rule.

· Characteristic
Values and Vectors: Trace and Determinant, Symmetric Matrices, Spectral
Decomposition, Quadratic Forms & Characterization.

II. Real Analysis.

· Basic
concepts: Norm, Distance, Open & Closed Sets, Convergence of Sequences, Compact
Sets, Continuous functions, Existence of solutions to constrained optimization
problems.

· Differential
calculus: Partial derivatives, chain rule, Homogenous functions and Euler’s
theorem, Inverse and Implicit function theorems

· Convex
analysis: Convex sets, Separating hyperplane theorem, Continuous and
Differentiable functions on convex sets, Concave functions, Quasi-concave
functions.

· Set
valued mappings & Continuity. Maximum theorem and Fixed point
theorems.

III. Classical Optimization Theory.

· Unconstrained
optimization: Necessary and sufficient conditions for Local and Global Maximum,
the Method of Least Squares and the Envelope Theorem.

· Constrained
Optimization: Necessary conditions for a local maximum, the Arithmetic
Mean-Geometric Mean inequality, Sufficient conditions for a local and a global
maximum.

IV. Modern Optimization Theory.

· Concave
programming: Constrained Global Maxima and Saddle Points, the Kuhn-Tucker
conditions, Constrained Local and Global Maxima.

· Quasi-concave
programming: the Sufficiency Theorem of Arrow-Enthoven

· The
Necessity Theorem of Arrow-Enthoven.

· Linear
Programming: the Primal and Dual problems, Optimality Criterion, the Basic
Duality Theorems, Complementary Slackness.

· Monotone
comparative statics.

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