Mathematical Economic Analysis
Instructor |
Teaching Assistant |
---|---|
Professor Alan G. Isaac |
Jeff Wheeble |
Office: Zoom (via Canvas) |
Office: Zoom (or arranged) |
Office Hours: M9-12; by appointment |
Hours: W 9:30am-11:30am; by appointment |
Email: aisaac@american.edu |
Email: jw2841a@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 online interactions, all students are expected to adhere to basic etiquette: be respectful, and quote appropriately.
Software
This course requires the use of Mathematica, both in the classroom and for homework assignments. Homework assignments must be typed in Mathematica. 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: 2023-10-19
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 crashes are not an excuse for late submission: be sure to save often and to back up to the cloud (e.g., with Google Drive or Drop Box). At the top of each assignment, include your name, a course identifier, and the assignment number.
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.
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. But, be sure to write up your homework assignments without assistance.
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 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 attentive class, I curve the grades. Specifically, if fewer than 15% of the class reaches the 90% cutoff, I will lower all cutoffs proportionally until at least 15% of the grades are A or A-.
Points
Points are earned as follows:
attendance (5 percent of points possible),
homework (20 percent of points possible),
midterm exam (30 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 several chapters of [Hoy.etal-2022-MITPress] and [Velleman-2019-CambridgeUP]. I strongly recommend acquiring these two texts, but they should also be available for Reserve Reading. You will also need to learn a bit of Mathematica for your homework assignments. Accordingly, the following books should prove especially helpful for this course.
- Mathematics for Economists
- How to Prove It: A Structured Approach 3rd Edition
[Velleman-2019-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-2017-WolframMedia] (free at https://www.wolfram.com/language/elementary-introduction/2nd-ed)
In addition, I have requested that some additional texts be put on reserve in the University Library. (Find a list on the syllabus supplement.) I require small amounts of reading from a few of these texts.
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)
- How to Prove It: A Structured Approach
[Velleman-2019-CambridgeUP] ch. 1–3
Strongly Recommended:
- Logic (in Applied Discrete Structures)
Exponents and Logarithms
Week 0
Review this material on your own. It will not be covered in class.
Review of Functions
Week 1–2
- Sets, Numbers, and Functions
[Hoy.etal-2022-MITPress] ch. 2
There are additional optional readings on functions.
Review of Set Operations
Week 3
- Sets, Numbers, and Functions
[Hoy.etal-2022-MITPress] ch. 2 (redux)
- Operations on Sets
[Velleman-2019-CambridgeUP] ch. 1.4
There are additional optional readings on sets.
Counting and Probability
Week 3-4
- Counting
[Hammack-2018-self] ch.3
- Discrete Probability Distributions
[Grinstead.Snell-2003-AMS] ch.1 https://open.umn.edu/opentextbooks/textbooks/21
- Combinatorics
[Grinstead.Snell-2003-AMS] ch.3 https://open.umn.edu/opentextbooks/textbooks/21
Strongly Recommended:
- Basic Probability Theory,
[Hansen-2022a-PrincetonUP] Chapter 1
- Random Variables
[Hansen-2022a-PrincetonUP] Chapter 2
- Parametric Distributions
[Hansen-2022a-PrincetonUP] Chapter 3
Binary Relations
Week 5–6
- How to Prove It: A Structured Approach, ch.4
There are additional optional readings.
Advanced Reading
- Collective Choice and Social Welfare, ch. 1*
- Relations
[Ok-2007-PrincetonUP] ch. A.1.2, A.1.3, A.1.4
- A Test of Consumer Demand Theory Using Observations of Individual Consumer Purchases
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
Recommended:
- An Introduction to Mathematical Analysis for Economic Theory and Econometrics, ch.3, 4.1
There are additional optional readings for this topic.
Sequences
Week 7
- Sequences, Series, and Limits
[Hoy.etal-2022-MITPress] ch. 3
- Mathematics for Economists, ch.12 (Sequences) and 23 (Difference Equations)
- Fixed Points
Recommended:
- An Introduction to Mathematical Analysis for Economic Theory and Econometrics, ch. 4
Midterm
Scheduled for 2023-10-19. (Does not cover Taylor series.)
Function Limits and Continuity
On your own, review the material on univariate differential calculus from your previous calculus course.
Week 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
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.
- 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
Eigensystems
Week 13
- Eigensystems
[Hoy.etal-2022-MITPress] ch. 10.1
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
- 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
References
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, Dean, Maxwell B. Stinchcombe, and Juraj Zeman. (2009) An Introduction to Mathematical Analysis for Economic Theory and Econometrics. Princeton, NJ: Princeton University Press.
Doerr, Al, and Ken Levasseur. (2021) Applied Discrete Structures. : Lulu.
Grinstead, Charles M., and J. Laurie Snell. (2003) Introduction to Probability. : American Mathematical Society. https://open.umn.edu/opentextbooks/textbooks/21
Hammack, Richard. (2018) Book of Proof. Richmond, VA: self published. https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf
Hansen, Bruce E. (2022) Probability and Statistics for Economists. Princeton, NJ: Princeton University Press. https://www.ssc.wisc.edu/~bhansen/probability
Hoy, Michael, et al. (2022) Mathematics for Economists. Cambridge, MA: MIT Press.
James, Gareth, et al. (2023) An Introduction to Statistical Thinking: With Applications in Python. New York, NY: Springer.
Langtangen, Hans Petter. (2009) A Primer on Scientific Programming with Python. : Springer.
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, 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
Ok, Efe A. (2007) Real Analysis with Economic Applications. Princeton, New Jersey: Princeton University Press.
Sen, Amartya. (1970) Collective Choice and Social Welfare. San Francisco: Holden-Day.
Simon, Carl P., and Lawrence Blume. (1994) Mathematics for Economists. New York: W.W. Norton & Company, Inc..
Stachurski, John. (2022) Economic Dynamics: Theory and Computation. Cambridge, MA: MIT Press.
Velleman, Daniel J. (2019) How to Prove It: A Structured Approach. Cambridge, UK: Cambridge University Press.
Walker, Mark. (1977) On the Existence of Maximal Elements. Journal of Economic Theory 16, 470--474.
Wolfram, Stephen. (2017) An Elementary Introduction to the Wolfram Language. Champaign, IL: Wolfram Media. https://www.wolfram.com/language/elementary-introduction/2nd-ed/
The syllabus above is Copyright © 2023 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).
Required and Recommended Syllabus Sections
The following sections are required or recommended on all syllabi at American University. The language is unaltered from suggestions provided by the administration.
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