## Class Times

The class meets on Tuesday and Thursday from 5:00pm to 6:15pm in DM-164.

## Cross-listing

This course is cross-listed as 6405/7405. Since they meet concurrently in the same room, the material covered will be the same.

## Course Description

This course focuses on mathematical methods used in modern economics. These include:

- linear methods often used in mathematical modelling,
- the portions of mathematical analysis relevant for studying optimization problems,
- the construction and solution of optimization problems, and
- the solution of difference and differential equations characteristic of modern intertemporal models, as used in both microeconomics and macroeconomics.

The first part of the course develops some basic mathematical tools of analysis which we will use to solve optimization problems. This covers roughly parts II and III of the text, and may include excerpts from parts VI and VII. The second part (part IV of the textbook) covers classical, calculus-based methods of optimization including Lagrange multipliers and the Kuhn-Tucker theorem. The methods of Lagrange and Kuhn-Tucker have been invaluable in solving many of the problems you will typically encounter in economics (consumer and producer choice, social welfare max, etc.). We then cover the solution of difference and differential equations, and their stability properties (part V). If time permits, we will look at dynamic optimization and the Maximum Principle.

## Course Objectives

By the end of the course, at a minimum, you should be able to:

- Determine whether a linear system has a solution, and if so, how many.
- Solve linear systems using both determinants and the Gauss-Jordan method.
- Find eigenvalues and eigenvectors.
- Use the functional calculus.
- Determine whether an optimization problem has a solution.
- Characterize the solutions of optimization problems via the first order conditions.
- Solve unconstrained optimization problems using first and second order conditions.
- Solve constrained optimization problems using the Kuhn-Tucker Theorem.
- Exploit special features such as homogeneity or convexity when solving optimization problems.
- Solve linear difference and differential systems.
- Characterize the long-run behavior of difference and differential systems using eigenvalues.

## Textbook, Slides, and Optimization Handout

### Textbook

- Carl Simon and Lawrence Blume,
*Mathematics for Economists*, W. W. Norton, New York, 1994.

Simon and Blume's book is the main text. I plan to cover Parts II-IV and VII of Simon and Blume, with some excerpts from Part VI. Time permitting, we will then turn our attention to Part V and dynamic models.

### Slides

- Chapters 6 and 7.
- Chapters 8 and 9.
- Chapter 10.
- Chapter 11+.
- Chapter 12+.
- Chapter 13, 29+.
- Chapter 14.
- Chapter 30, 29.3.
- Chapter 15.
- Chapter 16+.
- Chapter 17.
- Chapter 18.
- Chapter 19.
- Chapter 20.
- Chapter 21.

### Optimization Handout

You may find the following handout on basic optimization helpful, particularly in your micro course: Constrained Optimization Survival Guide.

## Selected Mathematical Economics Books

The first group focuses on mathematical economics. The following have been widely used and I am familiar enough with them to comment.

- Alpha Chaing and Kevin Wainwright,
*Fundamentals of Mathematical Economics*

Now in its 4th edition, this book is easier than Simon & Blume. I haven't seen this edition. I gather it adds material on probability and optimal control. Compared to S&B, it focuses more on how to use techniques rather than mathematical rigor. - Avinash K. Dixit,
*Optimization in Economic Theory*

A nice short book on both static and dynamic optimization. - Angel de la Fuente,
*Mathematical Methods and Models for Economists*

This book is more advanced than S&B and includes material on correspondences and fixed point theorems. - Rangarajan K. Sundaram,
*A First Course in Optimization Theory*

Raghu's book focuses on optimization. It's at a higher level than Dixit. - Akira Takayama,
*Mathematical Economics*

Akira's book is certainly mathematical, but the focus is on microeconomics, including general equilibrium and optimal growth. It's not really suitable for this course.

### Selected General Mathematics Books

The book by Garrity covers a fair chunk of the math we cover, plus quite a lot we don't use. I found its preface outstanding. The other two are on how to solve mathematical problems, especially those involving proofs.

- Thomas A. Garrity (2021),
*All the Math Your Missed*, 2nd ed., Cambridge University Press, Cambridge UK.

This is a book aimed at beginning graduate math students. There are many relatively elementary topics that students may encounter in their undergraduate classes, that are nonetheless assumed known when they show up in grad school. Surprise, surprise, much of that material is needed in our class too. While I cover some of it, I don't time to do everything. Even if you don't read any of the rest, use Amazon's Look Inside feature to read the excellent preface and other materials prior to chapter one. - George Polya (1957),
*How to Solve It: A New Aspect of Mathematical Method*, 2nd ed., Princeton University Press, Princeton, NJ.

If you're having difficulty understanding the logic of proofs, or how to find and construct them, try this classic. - Daniel J. Velleman (2019),
*How to Prove It: A Structured Approach*, 3rd ed., Cambridge University Press, Cambridge UK.

This is another take on the question of how to prove things.

### Two Free Math Books

There are a couple of decent math books available for free.

- For Linear Algebra, there's Jim Hefferon's free book, Linear Algebra.
- For basic (point set) topology, try Sidney A. Morris's Topology without Tears, which now includes some supplementary videos and translations of earlier versions into 8 languages (Arabic, Chinese, Greek, Korean, Persian, Russian, Spanish, and Turkish).

## Office Hours and Contact Info

If you have questions, you may ask immediately after class, or come
to my office.
Regular office hours are 12:45-1:45pm and 3:30-4:15pm on Tuesdays and Thursdays.
I will be happy to make an appointment for another time if that
is more convenient. My office is **DM-311A**,
my phone number is **305-348-3287**,
and my email is <boydj@fiu.edu>
or <John.Boyd@fiu.edu>.

## Exams and Homework

Grades will be based on two midterm exams (worth 25% each), a final exam (40%), and homework assignments (10%). In addition to being announced in class, homework assignments will be posted below.

Homework will be submitted in person or by emailing it to me. If you
email it, it may be easiest to write it out and then photograph it with
your phone. If so, please combine the pages into a **single pdf**.
I will not be happy if I see 10 separate files for one assignment.

Homework is graded as follows: ✓+ (3 pts) means that it is mostly correct, no major errors. ✓ (2 points) indicates you've missed at least one problem. ✓- (1 point) means that at least two problems or equivalent are mostly incorrect. On difficult assignments three misses may be required for a ✓-. A zero is also possible, and usually means it wasn't turned in.

### Homework Assignments and Answers

Assignments will appear here. Answers will be posted sometime after the homework is collected.

- Problems 6.1, 6.6, 7.7, 7.22, and 7.25 were due on Tuesday, September 6. Here are the answers.
- Problems 8.3, 8.18, 8.29, 9.8, and 9.13 were due on Tuesday, September 13. Here are the answers.
- Problems 12.15, 12.16, 12.20, 12.21, and 13.17 were due on Tuesday, October 11. Here are the answers.
- Problems 29.3, 29.9, 29.11, and 29.13 were due on Tuesday, October 18. Here are the answers.
- Problems 14.2, 14.4, 14.8, and 14.28 were due on Tuesday, October 25. Here are the answers.
- Problems 15.17, 15.38, 16.1, 16.6, and 17.9 were due on Tuesday, November 15. Here are the answers.
- Problems 18.2, 18.7, 18.13, 19.2, and 19.3 were due on Tuesday, November 29. Here are the answers.

### Exams

There will be two midterm exams, each worth 25% of your grade, and a final, worth 40% of your grade.

- The first midterm was on
**Thursday, September 22**. Click here for answers. - The second midterm was on
**Thursday, October 27**. Click here for answers. Click here for the extra credit questions and answers. - The final will be at the officially scheduled time,
**5pm on Tuesday, December 6, 2022**in our regular classroom, DM-164.

### Sample Exams

The material covered varies from year to year and some of the questions on previous exams may not be relevant for the material we cover this year. A few of the answers contain minor errors.

## Course Outline

Tentative and subject to change, especially if there are hurricanes. It is probably a bit over ambitious, but we'll see. There have been years when we've made it all the way to the end. Once the semester is underway, the slides will show what was actually covered.

Aug. 23 | ✓ 6: Intro to Linear Algebra (and use in Economics) |

Aug. 23, 25 | ✓ 7: Linear Systems |

Aug. 30 | ✓ 8: Matrix Algebra |

Sept. 1 | ✓ 9: Determinants & 26: Determinants |

Sept. 6, 8 | ✓ 10: Euclidean Spaces |

Sept. 13, 15 | ✓ 11: Linear Independence, Bases (see also Chapters 27 & 28) |

Sept. 20 | ✓ Change of Basis, Counting |

Sept. 22 | ✓ Exam #1 — through Chapter 11 + parts of 26, 27, and 28 |

Sept. 27 | No Class. FIU closed for Hurricane Ian |

Sept. 29 | ✓ 12: Limits and Closed Sets |

Oct. 4 | ✓ 12: Limits and Closed Sets |

Oct. 6 | ✓ 13.4: Continuous Functions |

Oct. 11, 13 | ✓ 29.1-2, 5: Monotone Convergence, Completeness, Compact Sets
✓ 30.1: Weierstrass Theorem |

Oct. 18 | ✓ 14: Calculus of Several Variables I |

Oct. 20 | ✓ 30: Calculus of Several Variables II: Rolle's Theorem,
✓ Mean Value Theorem, Taylor Formulas |

Oct. 25 | ✓ 29.3: Connected Sets, Intermediate Value Theorem |

Oct. 27 | ✓ Exam #2 — Chapters 12-14, 29 & 30 |

Nov. 1 | ✓ 15: Implicit Functions and their Derivatives |

Nov. 3 | ✓ 15: Implicit Functions and their Derivatives
✓ 16: Quadratic Forms and Definite Matrices |

Nov. 8 | ✓ 16: Quadratic Forms and Definite Matrices |

Nov. 10 | ✓ 17: Unconstrained Optimization
✓ 18: Constrained Optimization I: First-order Conditions |

Nov. 15 | No Class. Illness |

Nov. 17 | ✓ 18: Constrained Optimization I: First-order Conditions |

Nov. 22 | ✓ 19: Constrained Optimization II: Multipliers and Second-order Conditions |

Nov. 24 | Thanksgiving Holiday (no class) |

Nov. 29 | ✓ 20: Homogeneous and Homothetic Functions |

Dec. 1 | ✓ 21: Concave and Quasiconcave Functions |

Dec. 6 | Final Exam: At 5pm in our regular classroom, DM-164 |