Introductory video • Postscript to introductory video

This course covers mathematical methods commonly used in economic theory. In conjunction with MAT223 (Linear Algebra I), it is designed to be sufficient preparation for third- and fourth-year economics courses.

The topics covered are multivariate calculus (focusing on the tools used in economics), concavity and convexity, constrained optimization, and differential equations. Illustrative examples are taken from economics, but the purpose of the course is to teach mathematical methods, not economic theory.

The main aim of the course is to teach you the techniques commonly used to solve the mathematical problems that arise in economics. A secondary aim is to teach you how to make rigorous mathematical arguments. The ability to make such arguments deepens your understanding of the techniques and also allows you to modify the techniques when they do not exactly fit a problem you have to solve. With the second aim in mind, I will lead you through some proofs.

The course content is sufficient mathematical preparation for a Master's degree in Economics. If you plan to continue to a PhD, you should consider taking more rigorous courses, like MAT235 or MAT237 and higher-level math courses. (These courses do not cover all the topics in this course, but are at a higher theoretical level.) Alternatively, you could take this course and take higher-level math courses later.

#### Text

All the material for the course is covered in an online book designed specifically for this course. (This resource bills itself as a "tutorial", but to avoid confusion with the weekly tutorials for the course, I call it a "book".) This book is intended to be self-contained, but if you wish to refer to another book, my favorite is *Mathematics for economic analysis* by Knut Sydsæter and Peter J. Hammond (Prentice Hall, 1995). Unfortunately this book is out of print. (The authors have written another related book, *Essential mathematics for economic analysis*, which does not fit the course as well (and is expensive).) If you are relatively comfortable with the material you could look at a somewhat more advanced book, *Mathematics for economists* by Carl P. Simon and Lawrence Blume (Norton, 1994). This book is a bit more advanced than the course, but if you are comfortable with a formal approach you might like it.

#### Prerequisites and exclusions

The prerequisites for the course are ECO100Y1(67%)/( ECO101H1(63%), ECO102H1(63%))/ ECO105Y1(80%); MAT133Y1(63%)/( MAT135H1(60%), MAT136H1(60%))/ MAT137Y1(55%)/ MAT157Y1(55%), the corequisites are ECO200Y1/ ECO204Y1/ ECO206Y1, and the exclusions are MAT235Y1, MAT237Y1.

I expect you to be familiar with basic mathematical concepts and the following topics, which are covered in the prerequisite courses.

- Basic logic.
- Matrices and solutions of simultaneous linear equations (including determinants and Cramer's rule).
*Note*:**If you have not studied matrices previously, you need to do so before taking this course.**(You can either learn the material independently, or take a basic math course that covers them.) - One variable calculus (differentiation and integration, including exponential and logarithmic functions).
- Basic multivariate calculus (partial differentiation).
- Curve and set sketching.
- Basic optimization for functions of a single variable (finding maxima and minima using calculus).

The first five topics are covered briefly in the first section of the online book; I will briefly review them in the first class. To check your knowledge, you should do all the exercises in the first section of the tutorial:

- Exercises on logic
- Exercises on matrices: determinants, inverses, and rank
- Exercises on solving systems of linear equations: Cramer's rule and matrix inversion
- Exercises on intervals and functions
- Exercises on calculus: one variable
- Exercises on calculus: many variables
- Exercises on graphical representation of functions.

*You are prepared for the course if and only if you have little or no difficulty with these exercises*.

I will cover the last topic with which you should be familiar (basic optimization) later in the course.

If you need to review the material, you can refer to the text used in the prerequisite course, or read the book by Sydsæter and Hammond, or consult the first section of the on-line book. The following sections of Sydsæter and Hammond are relevant.

- Material you should know, very little of which I will review:
- Chs. 1, 2, 3, 12, 13.

- Material you should know, some of which I will review:
- Ch. 4 (but only the idea, not the details, of limits)
- Ch. 5 except 5.4 (covered in the course), 5.5, and 5.6.
- Ch. 6 through 6.5 (6.1--6.3: basic ideas only)
- Section 7.1 (7.2 is covered in the course)
- Ch. 8 through 8.4
- Ch. 9 through 9.4
- Ch. 10
- Ch. 11 through 11.2
- Ch. 15 through 15.6