Mathematical Economic Analysis

Contact Information for Fall 2024

Instructor

Teaching Assistant

Professor Alan G. Isaac

Linh Khuat

Office: Zoom (via Canvas)

Office: Kreeger G02 (or Zoom by appt)

Office Hours: M9-12; by appointment

Hours: T 2:00-3:00; by appointment

Email: aisaac@american.edu

Email: lk3393a@american.edu

Econ 705 Overview

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.

Course Policies

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 and Grading

Assessment

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

Class Participation

Class participation is expected but is ungraded. Class participation includes both in-class participation and asynchronous participation in online discussion. Both are used for ongoing assessment of student mastery of this course's learning outcomes (listed above). Assessments based on in-class participation allow immediate feedback and spontaneous discussion, with the goal of enhancing mastery of student learning outcomes.

Exams

Exams are graded, and no collaboration is allowed on exams. Exams presume a thorough knowledge of the graded and ungraded homework assignments given throughout the semester. Unless an exception is announced, exams are taken without the aid of textbooks or of notes of any kind.

In this course, I offer no makeup exams. (If you miss an exam due to unavoidable contingencies, such as illness or family emergencies, please obtain an excused absence from the Dean of Students. Your grade will then be calculated from the remaining exams.) There is no “extra credit”.

MIDTERM EXAMINATION: see our Canvas Assignments page.

FINAL EXAMINATION: check the final exam schedule for the date of our final examination. (The official final exam schedule always determines the date and time of the final.)

Homework

Homework assignments must be submitted on time via Canvas. Computer problems are not an excuse for late submission: be sure to use autosave and to back up to the cloud.

The TA for this course grades the homework. You may request supplementary comments from the TA, but do not request grade changes. The TA is not authorized to make grade changes. If you wish to contest a homework grade, you may submit your homework to me for regrading of the entire assignment. Be aware that I instruct my TA to be quite generous in grading, and regrading may produce a score reduction.

Homework assignments may include problems designated as optional. These are not collected or graded, but they can help you practice for subsequent classes and exams.

At the top of each assignment, include your name, a course identifier, and the assignment number. Ongoing study groups are highly recommended. Study groups are an excellent means of mastering the course material. They are also a core part of the experience of graduate education. Besides, they are fun.

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.

Points

Points are earned as follows:

  • homework (20 percent of points possible),

  • midterm exam (35 percent of points possible),

  • cumulative final exam (45 percent of points possible).

Topics and Readings

This is not a textbook based course. However, it draws fairly directly on many chapters of [Hoy.etal-2022-MITPress] and core chapters of [Fuchs-2023-CambridgeUP]. I strongly recommend purchasing these two texts, but I also asked that they be available for Reserve Reading. You will also need to learn a bit of the Wolfram Language for your homework assignments. Accordingly, the following books should prove especially helpful for this course.

Mathematics for Economists

[Hoy.etal-2022-MITPress]

Introduction to Proofs and Proof Strategies

[Fuchs-2023-CambridgeUP] (The Kindle version is inexpensive and may be read and annotated on computer with the Kindle app.)

An Elementary Introduction to the Wolfram Language

[Wolfram-2024-WolframMedia] (free at https://www.wolfram.com/language/elementary-introduction/3rd-ed/)

In addition, I have requested that some additional texts be put on reserve in the University Library. I require a small amount of reading from a few of these texts, as specified on this syllabus and in Canvas.

The suggested timing of topics and extent coverage is tentative and will be revised as the semester progresses. The number of classes designated below for each topic is intended only as a rough guide: class interest and preparation will determine how quickly we progress through the topics. New readings may be added to the readings during the course. Readings listed as recommended are entirely optional.

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)

Advanced Reading

Collective Choice and Social Welfare, ch. 1*

[Sen-1970-HoldenDay]

A Test of Consumer Demand Theory Using Observations of Individual Consumer Purchases

[Battalio.etal-1973-WEJ]

There are additional optional readings.

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.

Linear Equations and Vector Spaces

[Hoy.etal-2022-MITPress] ch. 7

Elementary Equation Operations

Week 10

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

We may not have time to discuss chapter 14.

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

Optional: Advanced Reading in Multivariate Optimization

More Constrained Multivariate Optimization

[Hoy.etal-2022-MITPress] ch. 13

Envelope Theorem

[Hoy.etal-2022-MITPress] ch. 14.3

Kuhn-Tucker Conditions

[Hoy.etal-2022-MITPress] ch. 15

If Time Remains

Markov Chains

Economic Dynamics: Theory and Computation, ch. 4

[Stachurski-2022-MITPress]

Dynamic Optimization

Economic Dynamics: Theory and Computation, ch. 5

[Stachurski-2022-MITPress]

Differential Equations

handout

Recommended:

[Simon.Blume-1994-WWNorton] ch.23, 24, 25;

References

[Battalio.etal-1973-WEJ]

Battalio, Raymond C., et al. (1973) A Test of Consumer Demand Theory Using Observations of Individual Consumer Purchases. Western Economic Journal 11, 411--428.

[Corbae.Stinchcombe.Zeman-2009-PrincetonUP]

Corbae, Dean, Maxwell B. Stinchcombe, and Juraj Zeman. (2009) An Introduction to Mathematical Analysis for Economic Theory and Econometrics. Princeton, NJ: Princeton University Press.

[Doerr.Levasseur-2021-Lulu]

Doerr, Al, and Ken Levasseur. (2021) Applied Discrete Structures. : Lulu.

[Fuchs-2023-CambridgeUP]

Fuchs, Shay. (2023) Introduction to Proofs and Proof Strategies. : Cambridge University Press.

[Grinstead.Snell-2003-AMS]

Grinstead, Charles M., and J. Laurie Snell. (2003) Introduction to Probability. : American Mathematical Society. https://open.umn.edu/opentextbooks/textbooks/21

[Hammack-2018-self]

Hammack, Richard. (2018) Book of Proof. Richmond, VA: self published. https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf

[Hansen-2022a-PrincetonUP]

Hansen, Bruce E. (2022) Probability and Statistics for Economists. Princeton, NJ: Princeton University Press. https://www.ssc.wisc.edu/~bhansen/probability

[Hoy.etal-2022-MITPress]

Hoy, Michael, et al. (2022) Mathematics for Economists. Cambridge, MA: MIT Press.

[James.etal-2023-Springer]

James, Gareth, et al. (2023) An Introduction to Statistical Thinking: With Applications in Python. New York, NY: Springer.

[Langtangen-2009-Springer]

Langtangen, Hans Petter. (2009) A Primer on Scientific Programming with Python. : Springer.

[Lukianoff.Haidt-2018-Penguin]

Lukianoff, Greg, and Jonathan Haidt. (2018) The Coddling of the American Mind: How Good Intentions and Bad Ideas Are Setting Up a Generation for Failure. London, England: Penguin Books.

[May.Elder-2018-IJEdTech]

May, Kaitlyn E., and Anastasia D. Elder. (2018) Efficient, Helpful, or Distracting? A Literature Review of Media Multitasking in Relation to Academic Performance. International Journal of Educational Technology in Higher Education 15, Article 13. https://doi.org/10.1186/s41239-018-0096-z

[Sen-1970-HoldenDay]

Sen, Amartya. (1970) Collective Choice and Social Welfare. San Francisco: Holden-Day.

[Simon.Blume-1994-WWNorton]

Simon, Carl P., and Lawrence Blume. (1994) Mathematics for Economists. New York: W.W. Norton & Company, Inc..

[Stachurski-2022-MITPress]

Stachurski, John. (2022) Economic Dynamics: Theory and Computation. Cambridge, MA: MIT Press.

[Velleman-2019-CambridgeUP]

Velleman, Daniel J. (2019) How to Prove It: A Structured Approach. Cambridge, UK: Cambridge University Press.

[Walker-1977-JET]

Walker, Mark. (1977) On the Existence of Maximal Elements. Journal of Economic Theory 16, 470--474.

[Wolfram-2024-WolframMedia]

Wolfram, Stephen. (2024) An Elementary Introduction to the Wolfram Language. Champaign, IL: Wolfram Media. https://www.wolfram.com/language/elementary-introduction/2nd-ed/

Version: 2024-10-23

The syllabus above is Copyright © 2024 by Alan G. Isaac. Some rights are reserved. This work is licensed under the Creative Commons Attribution-ShareAlike License version 2.0 (or any subsequent version).