Mathematical Economic Analysis

Learning Outcomes

This course is an introduction to mathematical methods used by economists. These are basic mathematical tools needed for advanced study in economics. Students who master the material will be able to understand and solve problems related to:

• types of functions in economic models.

• composition of functions.

• sets and methods of proof.

• linear models and matrix algebra.

• comparative static analysis of linear models.

• differential calculus.

• comparative static analysis of non-linear models.

• univariate and multivariate optimization.

• comparative dynamic analysis.

• simple computer applications of these mathematical methods.

Every class differs in background preparation and interests. I will depend on your classroom feedback to set the class pace and the depth of coverage of certain topics, so be sure to ask questions in class.

Course Prerequisites

Regarding the calculus prerequisite: I assume you remember basic differential calculus from the first semester and a little bit about integration. Ideally you also recall some statistics and matrix algebra, but we will start from scratch with these topics.

This course is taught at a level appropriate to highly committed first-year PhD students. In past years I had to give some failing grades (C, D, and F) in this course, which is very unpleasant for both instructor and student. Please ensure that you are fully committed to attempting this very demanding course.

Communication

This class will use Canvas. Look there for the syllabus, lecture supplements, and assignments. Canvas announcements are sent by email; students must monitor these announcements and Canvas Conversations (Inbox). Students should also subscribe to the Canvas Discussions. In any online discussion, be sure to adhere to basic etiquette: be respectful, and quote appropriately.

Classroom Policies (IRW)

The following policies apply to in-person meetings only.

Masks:

Be sure to follow the AU mask guidelines. These are updated on a regular basis, in response to local conditions. Use a high-quality mask (e.g., KN95). Surgical masks are not adequate. When wearing a mask, ensure that it fits correctly. The mask must be snug over both the nose and mouth.

Connected devices:

As a courtesy to other students, do not browse the web or check your email during class. Such activities distract others and reduce your ability to contribute to discussions. In addition, there is evidence that students believe themselves capable of productive multitasking but in fact are not (May and Elder, 2018).

Cell phones:

Please silence your phone before entering the classroom. (E.g., put it on vibrate, or turn it off.) Place it out of sight in a pocket or bag (and not on your desk). If you need to make or take a call, you may quietly leave the classroom. (Please sit near the door if you anticipate such a need.)

Computers:

You may use a computer to take notes or do other course-relevant activity. However, be aware that research indicates that using a computer to take notes can hinder learning (Sana, Weston, Cepeda 2013). This appears to be due to the distraction of multitasking.

Software

This course requires the use of Mathematica, both in the classroom and for homework assignments. Homework assignments must be typed into a Mathematica notebook. You will produce your homework as Mathematica notebook (.nb) files. However, your homework submission must be a PDF file. To submit, first save the notebook, and then save it again in PDF format. Finally, upload the PDF file on Canvas.

You are required to turn on autosave and additionally to use an external backup such as GoogleDrive for Desktop or OneDrive. Lost work is not an excuse for late submission!

American University has been providing free Student Version Licenses of Mathematica. Nevertheless, before leaving the university, consider purchasing Mathematica at the very attractive student price.

Assessment

Mastery of the student learning outcomes (see above) is assessed by means of graded exams, graded homeworks, and ungraded class participation.

Grading

Homework and exams are graded. Grades are based on the total points earned. (See below.) There is no "extra credit". Whole grade numerical cut-off points are fairly standard: 90% for A-range grades and 80% for B-range grades. It follows that in principle everyone can attain an A, if every student makes an exceptional effort. In practice, this has not happened.

If numeric scores are unusually low in an otherwise performant class, I curve the grades. Specifically, if fewer than 15% of the class reaches the 90% cutoff, I typically lower all cutoffs proportionally until approximately 15% of the grades are A- or higher. Usually, this adjustment is not needed.

Logic and Proofs

Week 0 (read before class starts)

Review this material on your own. It will not be covered in class.

Introduction to Proofs and Proof Strategies

[Fuchs-2023-CambridgeUP] ch. 1–3

Strongly Recommended:

How to Prove It: A Structured Approach

[Velleman-2019-CambridgeUP] ch. 1–3

Logic

[Hammack-2018-self] ch.2

Logic (in Applied Discrete Structures)

[Doerr.Levasseur-2021-Lulu] ch. 3

Numbers

Week 0 (read before class starts)

Integers and Divisibility

[Fuchs-2023-CambridgeUP] ch. 6

Exponents and Logarithms

Week 0 (read before class starts)

Review this material on your own. It will not be covered in class.

Functional Forms

[Hoy.etal-2022-MITPress] ch. 2.4.

Review of Functions

Week 1–2

Sets, Numbers, and Functions

[Hoy.etal-2022-MITPress] ch. 2

Mathematical Induction

[Fuchs-2023-CambridgeUP] ch. 4

There are additional optional readings for this review of functions.

Review of Set Operations

Week 3

Sets, Numbers, and Functions

[Hoy.etal-2022-MITPress] ch. 2 (redux)

Sets, Functions, and the Field Axioms

[Fuchs-2023-CambridgeUP] ch. 2 (redux)

There are additional optional readings for this review of set operations.

Sets, Counting, and Combinatorics

Week 3

Elementary Combinatorics

[Fuchs-2023-CambridgeUP] ch. 8

Counting

[Hammack-2018-self] ch.3

Strongly Recommended:

Combinatorics

[Grinstead.Snell-2003-AMS] ch.3 https://open.umn.edu/opentextbooks/textbooks/21

Basic Probability Theory,

[Hansen-2022a-PrincetonUP] Chapter 1

Random Variables

[Hansen-2022a-PrincetonUP] Chapter 2

Parametric Distributions

[Hansen-2022a-PrincetonUP] Chapter 3

Basic Probability Concepts

Week 4

Discrete Probability Distributions

[Grinstead.Snell-2003-AMS] ch.1 https://open.umn.edu/opentextbooks/textbooks/21

Binary Relations

Week 5–6

Relations

[Fuchs-2023-CambridgeUP] ch. 7

Relations

[Velleman-2006]_ ch.4 and [Velleman-2019-CambridgeUP] ch.4 (note: the 2nd edition has the material on the transitive closure)

Introduction to Real Analysis

Week 6-7

Sequences, Series, and Limits

[Hoy.etal-2022-MITPress] ch. 3

Mathematics for Economists, ch.12 and and 29

[Simon.Blume-1994-WWNorton] ch.12 (Sequences) and 29 (Limits and Compact Sets)

On the Existence of Maximal Elements

[Walker-1977-JET]

Recommended:

An Introduction to Mathematical Analysis for Economic Theory and Econometrics, ch.3, 4.1

[Corbae.Stinchcombe.Zeman-2009-PrincetonUP] ch.2;

There are additional optional readings for this topic.

Midterm

Midterm approximately here. See Canvas Assignments for the date.

Sequences

Week 8

Sequences, Series, and Limits

[Hoy.etal-2022-MITPress] ch. 3

Mathematics for Economists, ch.12 (Sequences) and 23 (Difference Equations)

[Simon.Blume-1994-WWNorton]

Fixed Points

Video: https://www.youtube.com/watch?v=csInNn6pfT4&t=125s

Recommended:

An Introduction to Mathematical Analysis for Economic Theory and Econometrics, ch. 4

[Corbae.Stinchcombe.Zeman-2009-PrincetonUP]

Function Limits and Continuity

Week 8-9

Your may wish to review the material on univariate differential calculus from your previous calculus course.

Week 9

Limits and Continuity

[Fuchs-2023-CambridgeUP] ch. 9

Continuity of Functions

[Hoy.etal-2022-MITPress] ch. 4

Derivatives and Differentials (Univariate)

Week 9

Review our discussions from math camp and then read:

Univariate Derivatives, Differentials, Taylor Series

[Hoy.etal-2022-MITPress] ch. 5

Recommended:

Taylor Series

Video: https://www.youtube.com/watch?v=KoCk1krAX2M

Optimization (Univariate)

Week 9

Univariate Optimization

[Hoy.etal-2022-MITPress] ch. 6

Functions on Vector Spaces

Week 10

This material is review. It will not be covered in class.

Linear Algebra

[Fuchs-2023-CambridgeUP] ch. 11

Applied Discrete Structures

[Doerr.Levasseur-2021-Lulu] ch. 5 provides a brief elementary introduction to matrices.

Elementary Equation Operations

Week 10

Linear Equations and Vector Spaces

[Hoy.etal-2022-MITPress] ch. 7.1 and 7.2

Matrices

[Hoy.etal-2022-MITPress] ch. 8

Determinants and Inverses: Some Details

Week 11

Determinants and Inverses

[Hoy.etal-2022-MITPress] ch. 9

The Comparative Statics of Linear Models

Week 12

Determinants and Inverses

[Hoy.etal-2022-MITPress] ch. 9

Multivariate Calculus

Week 12

Multivariate Calculus

[Hoy.etal-2022-MITPress] ch. 11

Nonlinear Comparative Statics

Week 12

Comparative Statics

[Hoy.etal-2022-MITPress] ch. 14.1, 14.2

Eigensystems

Week 13

Eigensystems

[Hoy.etal-2022-MITPress] ch. 10.1

We may not have time to discuss eigensystems.

Quadratic Forms

Week 13

Quadratic Forms

[Hoy.etal-2022-MITPress] ch. 10.2

Multivariate Optimization

Week 14

Multivariate Optimization

[Hoy.etal-2022-MITPress] ch. 12

Constrained Multivariate Optimization

Week 14

FONCs for Constrained Multivariate Optimization

[Hoy.etal-2022-MITPress] ch. 13.1

Second-Order Conditions for Constrained Multivariate Optimization

[Hoy.etal-2022-MITPress] ch. 13.2

Academic Support and Access Center (ASAC) MGC 243, 202-885-3360.

All students may take advantage of the Academic Support and Access Center (ASAC) for individual academic skills, counseling, workshops, tutoring and writing assistance, as well as Supplemental Instruction. All services are free. The services include the Writing Center (first floor of Bender Library), which assists students with academic writing and assignments. The Math/Stat Lab (Myers Building, 202-885-3154) which provides mathematics and statistics tutoring. Additional content tutoring is also available in the ASAC’s Tutoring Lab.

Academic Integrity

Standards of academic conduct are set forth in the University’s Academic Integrity Code. By registering for this course, you have acknowledged your awareness of the Academic Integrity Code and your obligation to become familiar with your rights and responsibilities as defined by the code. Violations of the Academic Integrity Code will not be treated lightly, and disciplinary actions will be taken should violations occur. The standard sanction for violations is failure of the course.

Emergency Preparedness

In the event of a declared pandemic (influenza or other communicable disease), American University will implement a plan for meeting the needs of all members of the university community. Should the university be required to close for a period of time, we are committed to ensuring that all aspects of our educational programs will be delivered to our students. These may include altering and extending the duration of the traditional term schedule to complete essential instruction in the traditional format and/or use of distance instructional methods. Specific strategies will vary from class to class, depending on the format of the course and the timing of the emergency. Faculty will communicate class-specific information to students via AU e-mail and Blackboard. Students are responsible for checking their AU e-mail regularly and keeping themselves informed of emergencies. In the event of a declared pandemic or other emergency, students should refer to the AU Web site (american.edu/emergency) and the AU information line at (202) 885-1100 for general university-wide information, as well as contact their faculty and/or respective dean’s office for course and school/college-specific information.

Sharing of Course Content

Students are not permitted to make visual or audio recordings, including live streaming, of classroom lectures or any class related content, using any type of recording devices (e.g., smart phone, computer, digital recorder, etc.) unless prior permission from the instructor is obtained, and there are no objections from any of the students in the class. If permission is granted, personal use and sharing of recordings and any electronic copies of course materials (e.g., PowerPoints, formulas, lecture notes and any classroom discussions online or otherwise) is limited to the personal use of students registered in the course and for educational purposes only, even after the end of the course.

Exceptions will be made for students who present a signed Letter of Accommodation from the Academic Support and Access Center. See: How Do I Request Accommodations?

http://www.american.edu/ocl/asac/Accommodations.cfm

To supplement the classroom experience, lectures may be audio or video recorded by faculty and made available to students registered for this class. Faculty may record classroom lectures or discussions for pedagogical use, future student reference, or to meet the accommodation needs of students with a documented disability. These recordings are limited to personal use and may not be distributed (fileshare), sold, or posted on social media outlets without the written permission of faculty.

Unauthorized downloading, file sharing, distribution of any part of a recorded lecture or course materials, or using information for purposes other than the student’s own learning may be deemed a violation of American University’s Student Conduct Code and subject to disciplinary action (see Student Conduct Code VI. Prohibited Conduct).