## Class Times

The class meets on Tuesday and Thursday from 5:00pm to 6:15pm via *Zoom*.
Links to the *Zoom* class will appear on the course's Canvas page.

## Course Description

The most commonly used mathematical methods in economics relate to optimization problems, and this course focuses on methods of optimization.

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

- 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.

## Lecture Notes

- Chapters 6 and 7.
- Chapters 8, 9, and 26.
- Chapters 10, 11, and 27, together with some extra material.
- Counting & Chapter 12.
- Chapters 13.4 & 29.1, 2, 5.
- Chapters 14, & 30
- Chapter 15
- Chapter 16
- Chapter 17
- Chapter 18
- Chapter 19
- Chapter 20
- Chapters 23 & 24
- Chapters 25 & Optimal Growth

## Optimization Handout

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

## Office Hours and Contact Info

Due to the continuing coronavirus hazard, I will not have regular office hours. I will be available via Zoom immediately after class. Just speak up once class ends if you have a question. Otherwise, you may contact me via email at <boydj@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 by emailing it to me. 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.

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 4 and 8 from Chapter 6 and problems 6, 14, 20, and 24 from Chapter 7 were due on Thursday, September 10. Here are the answers.
- Problems 4, 24, and 28 from Chapter 8, problems 11 and 13 from Chapter 9 and problem 16 fron Chapter 26 were due on Tuesday, September 22. Here are the answers.
- Problems 2, 7, and 25 from Chapter 12, problem 16 from Chapter 13, and problems 4 and 13 from Chapter 29 were due on Tuesday, October 13. Here are the answers.
- Problems 6, 12, and 17 from Chapter 14, problems 7 and 24 from Chapter 15 and problem 12 from Chapter 29 were due on Tuesday, October 20. Here are the answers.
- Problems 22 and 36 from Chapter 15, problems 2 and 3 from Chapter 16 and problem 7 from Chapter 30 were due on Tuesday, October 27. Here are the answers.
- Problems 4 and 7 from Chapter 17 and problems 5, 9, 11, and 15 from Chapter 18 were due on Thursday, November 12. Here are the answers.
- Problems 14, 18 and 22 from Chapter 19 and problems 1, 11, and 18 from Chapter 20 were due on Tuesday, November 24. Here are the answers.
- Problems 15 and 21 from Chapter 23 and problems 13 and 17 from Chapter 24 were due on Tuesday, December 1. Here are the answers.

### Exams

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

- The first midterm was on
**Thursday, September 24**. Here are the answers. - The second midterm was on
**Thursday, October 29**. Here are the answers. - The final has be rescheduled to
**Thursday, Dec. 10**at**5pm**. It will be at-home exam.

### 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

Subject to change, as happened last year due to Hurricane Dorian.

Aug. 25 | 6: Intro to Linear Algebra (and use in Economics) |

Aug. 25, 27 | 7: Linear Systems |

Sept. 1 | 8: Matrix Algebra |

Sept. 3 | 9: Determinants & 26: Determinants |

Sept. 8, 10 | 10: Euclidean Spaces |

Sept. 15, 17 | 11: Linear Independence, Bases (see also Chapters 27 & 28) |

Sept. 22 | Counting, 12: Limits and Open Sets |

Sept. 24 | Exam #1 — through Chapter 11 + part of 26, 27, and 28 |

Sept. 29 | 12: Limits and Closed Sets |

Oct. 1 | 13.4 Continuous Functions, 29.1: Monotone Convergence |

Oct. 6 | 29.1-2, 5: Completeness, Compact Sets, 30.1: Weierstrass Theorem |

Oct. 8 | 14: Calculus of Several Variables I |

Oct. 13 | 30: Calculus of Several Variables II
Rolle's Theorem, Mean Value Theorem, Taylor Formulas 29.3: Connected Sets, Intermediate Value Theorem |

Oct. 15, 20 | 15: Implicit Functions and their Derivatives |

Oct. 20, 22 | 16: Quadratic Forms and Definite Matrices |

Oct. 27 | 17: Unconstrained Optimization |

Oct. 27 | 18: Constrained Optimization I: First-order Conditions |

Oct. 29 | Exam #2 — Chapters 12-17, 29 & 30 |

Nov. 3 | 18: Constrained Optimization I: First-order Conditions (continued) |

Nov. 5 | 19: Constrained Optimization II: Multipliers and Second-order Conditions |

Nov. 10 | 20: Homogeneous and Homothetic Functions |

Nov. 12 | 21: Concave and Quasiconcave Functions |

Nov. 17 | 23: Eigenvalues and Eigenvectors |

Nov. 19 | 23: Eigenvalues and Eigenvectors, Complex Solutions |

Nov. 24 | 24: Ordinary Differential Equations: Scalar Equations |

Nov. 26 | Thanksgiving Holiday (no class) |

Dec. 3 | 25: Ordinary Differential Equations: Systems of Equations |

Dec. 5 | Introduction to Control Theory |

Dec. 8 | Final Exam: Dec. 10 at 5pm |