# Advanced Mathematical Economics

## Main course materials

Please regularly check that you have the
latest version my lecture notes,
which were last updated at
2:05PM, Tuesday 16 of March.
You can see what changed using Adobe Acrobat Pro on the uCreate computers in
the library.
Choose `Tools -> Compare Documents`

, and select the PDFs containing
the old and new versions of the notes.

You can read the Assessment Guide and Practice Questions and Sample Solutions.

You can ask questions and get involved in discussions on the course's Piazza page.

You can watch lecture recordings by logging into Learn and then opening MediaHopper Replay.

You can browse last year's course materials.

## News

- 26th of March:
- The May exam will be online and open book.
- The exam will be 3 hours, plus 1 hour to upload your answers.
- Even though the exam is open book, I recommend you prepare as if it were closed book. Creativity involves making connections between seemingly unrelated ideas. Your mind cannot do this if it does not already contain the ideas.
- If you would like me to write a recommendation letter based on your May exam, I would encourage you to request an interview with me shortly after the exam. This will help me write the most favourable possible letter for you.

- 20th of January:
- December exam marks are now available on Learn Grade Centre.
- Sample solutions and commentary are now available.
- You can view your exam script in my office, and receive feedback.
- Piazza and office hours will continue throughout the year.
- If you scored less than 85, then you should consider taking the May exam. Students at all levels often improve their logic and proof writing skills substantially.
- I am happy to write recommendation letters for everyone -- there is no minimum mark.

- Study advice and announcements:
- The list of examinable topics is now available below.
- I recommend that students practice using the old exam questions and solutions in the assessment guide above.
- I recommend that students attempt to master the first six topology topics (up to and including continuity) before putting much time into the more advanced topology topics.
- I recommend that you form study groups to check each others answers and to help each other. It's important to learn from your mistakes, which can be hard to spot. And teaching your friends is a great way to learn. You can email me if you want help in finding a study group.
- I recommend that you use Piazza and office hours up to the May exam.
- If you would like a recommendation letter to pursue postgraduate study, then please email me.

- Week 10: whiteboard notes
- We finished C9, and did C11 (bonus).
- Homework: Practice Question 31, C60, C62, C64, C68, C72.

- Week 9: whiteboard notes
- We finished G, did 4.3, and started C9.
- Homework: Practice Question 29Ai (May 2018), C51, C59, C61, G1, G4

- Week 8: whiteboard notes
- We did C8, 2.3, 2.4, some of 3.2, and started G.
- Homework: 2.13, 3.6, C.52, C.53, C.56, C.58.

- Week 7: whiteboard notes
- We finished C7 and continued through 2.3.
- Homework: 2.11, C.42, C.43, C.45, C.46, C.47
- I substantially changed the second half of the proof of Theorem C.14 (i.e. that (CB(X, Y), d_inf) is complete, the bit we skipped) in the notes.

- Week 6: whiteboard notes
- We finished C6, and started C7 and 2.3.
- Homework: C.30, C.34 (indifference curves only), C.38, C.40, C.41.

- Week 5: whiteboard notes
- We did 2.2, C5, and started C6.
- Homework: 2.6, 2.7, C.24, C.25, C.27, C.29, C.35
- We have a new tutorial room: Patersons land G21.

- Week 4: whiteboard notes
- We did C4, started C5, did D (skipping quasi-convexity), 2.1.
- Homework: Please read the proof that concavity implies decreasing returns to scale, and do questions C.13, C.14, C.16, C.17, C.22, C.23. Please ensure you have the latest version of the notes.

- Week 3: whiteboard notes
- We did C2, C3, more of 2.1 and started D.
- Homework: Questions C2, C3, C7, C8, C9, but not C12 (it was harder than I thought!)

- Week 2: whiteboard notes
- We did C1, and started C2 and 2.1.
- Some students are having trouble signing up for Piazza. I just sent everyone a sign-up email. Sorry for the spam!
- Homework: Read Appendix E1 and E4, and do Question E1. You can submit via Learn, or on paper at the start of the next lecture.
- The tutorial room has changed again. Please check the timetable to make sure you go to the right place.
- I added some hints relevant to the homework, both in the section B6 and in the question E1.

- Week 1: whiteboard notes
- The rooms have changed. Please check the timetable to make sure you go to the right place.
- We did B1, B2, B3, B5, and started C1. Please read all of B1-B6. I recommend you watch some of the logic videos.
- Homework (optional this week only): B1-B14. Please ask questions on Piazza.
- There is no tutorial this week.

- September 12th: The first lecture will be at 4pm on September 17th. We still don't know where, because the room we had booked was too small. Please keep an eye on the course timetable.
- July 12th: If you are interested in taking this course, I recommend you read the preparation guide. You should also consider taking the GRE test soon.

## Overview

This course teaches some of the important mathematical tools used by economists. More importantly, the course's intensive structure with tutorials every week is designed to train students how to think like mathematicians. Specifically, how to use mathematical notation to write clearly, how to write proofs, how to find counter-examples to conjectures, how to transform complicated problems into simple and elegant problems, and how to think abstractly.

The course is available both to University of Edinburgh students (undergraduate, masters, PhD) and to Continuing Professional Development (Mathematical Economics) students who are not enrolled on any degree.

This course is primarily targeted at students who would like to prepare for post-graduate study in economics. Mathematics is essential for advanced study of economics, and many top MSc and PhD programmes require university-level training in mathematics for admission. In the past, students have also taken this course to prepare for study in other areas including mathematics, cognitive science, computer science, data science, and finance. On the other hand, admissions committees for MBAs and professionally-oriented finance degrees are unlikely to put a high value on this course.

I recommend that students follow my guide to prepare for the course which involves watching videos to refresh their high-school mathematics knowledge and learn a bit about logic; this is also a good opportunity to take the GRE test. The course draws on economics examples, so Economics 2 (or equivalent) is also required. Students who have already taken the three first-year undergraduate courses in mathematics (Introduction to Linear Algebra, Calculus and Its Applications, and Proofs and Problem Solving) are already well-prepared for post-graduate study, although might still benefit from this course. Joint honours students with mathematics are welcome to take this class, although I recommend they "spend" their two economics options on courses more focused on social problems.

The main reference is my lecture notes, which I am updating regularly. You can download the Latex source if you want to annotate or contribute improvements to the notes.

Half of every lecture will be on the language of mathematics and metric spaces. The other halves will be on calculus, convex analysis, and dynamic programming.

## Extra Reference

Some students like an extra reference, although it is unnecessary. A clickable reading list is available with the same books as below, via the library.

Half of every lecture will be on the language of mathematics and metric spaces. The closest book to my notes is Rosenlicht's (1968) "Introduction to Analysis". I recommend that everyone buy a copy of Rosenlicht's book.

For the calculus and convex analysis topics, the closest book to my notes is Boyd and Vandenberghe's (2004) "Convex Optimization". For the dynamic programming topic, the closest book is Stokey and Lucas' (1989) "Recursive Methods in Economic Dynamics".

You might also find these books helpful: Kolmogorov and Fomin's (1970) "Introductory Real Analysis", Angel de la Fuente's (2000) "Mathematical Methods and Models for Economists", and Luenberger's (1969) "Optimization by Vector Space Methods".

A large part of the class is about writing proofs. This is an art in itself, and there are several books about this:

- Daepp and Gorkin's (2011) "Reading, Writing, and Proving: A Closer Look at Mathematics",
- Kane's (2016) "Writing Proofs in Analysis",
- Liebeck's (2015) "A Concise Introduction to Pure Mathematics",
- Oliveira and Stewart's (2015) "Building Proofs: A Practical Guide",
- Robert's (2010) "Introduction to Mathematical Proofs: A Transition",
- Solow's (2005) "How to read and do proofs: an introduction to mathematical thought process",
- Sundstrom's (2013) "Mathematical Reasoning: Writing and Proof" (open access),
- Velleman's (2006) "How to prove it: a structured approach".

The economics topics in my notes are closer to Varian and Kreps than MWG, but quite different from all of them.

MWG means Mas-Colell, Whinston and Green's (1995) "Microeconomic Theory". V means Varian's (1992) "Microeconomic Analysis". K means Kreps' (1990) "A Course in Microeconomic Theory". KK means Kreps' (2013) "Microeconomic Foundations 1: Choice and Competitive Markets". SL means Stokey and Lucas (1989), "Recursive Methods in Economic Dynamics". Debreu (1960) is Topological methods in cardinal utility theory.

- Introduction
- Production
- Production Functions
*See: V1, MWG5, K7.1* - Profit Maximization
*See: V2, MWG5, K7.2* - Upper Envelopes and Value Functions
*See: V3, SL4, MWG5, K7.2* - Cost Functions and Dynamic Programming
*See: V4, SL4, MWG5, K7.3, K.A.2* - Upper Envelopes with Constraints
*See: V5, SL4, MWG5, K7.3*

- Production Functions
- Consumption
- Utility Functions
*See: V7, MWG3, K2.1* - Time Preference
*See: Debreu (1960), V19, SL4, SL5, MWG20, KK2.5* - Utility Maximization
*See: V7, MWG3, K2.2* - Consumerâ€™s Value and Policy Functions
*See: V7, MWG3, K2.2, K2.3* - Expenditure Function and Policy Functions
*See: V7, MWG3, K2.2, K2.3* - Slutsky Decomposition
*See: V8, MWG3, K2.3*

- Utility Functions
- Equilibrium
- Economies
*See V17, V18, V19, MWG16, K6.1, K6.2* - Efficient Allocations
*See V17, V18, V19, MWG16, MWG22, K5.2* - Equilibrium
*See V17, V18, V19, MWG16, K6.1* - Characterising Equilibria
*See V17, MWG15, MWG16, MWG20, K2.2* - Efficiency of Equilibria
*See V17, MWG16, K6.3* - *Existence of Equilibria
*See V17, MWG17, K6.4* - Implementation of Efficient Allocations
*See V17, MWG16, K6.3*

- Economies

## Visiting student project

One-semester visiting students can do an one optional project due in December 20 at 11pm via email; you are welcome to send me drafts for comments.

## Examinable Topics

- Chapter 2 (Production) except the following:
- Quasi-concavity and upper contour sets.
- The constrained envelope theorem. (Section 2.5)
- Production technology sets. (Section 2.6)

- None of Chapter 3 (Consumption) is examinable except for the cake-eating problem.
- Section 4.3 (Pure-Exchange Equilibrium) as well as formulating competitive equilibrium models with firms. See the practice questions.
- The content of Appendix B (Naive set theory) will not be examined directly. However, it is the language of mathematics and economics, so you should be familiar with all of it (except the section on cardinality).
- Appendix C (Topology), except for C10 (connected sets), and the open-cover approach to compactness. This means that Cantor's intersection theorem is examinable, but not the Heine-Borel theorem. Exam questions might ask you to apply topology ideas to simple economic problems (like the Extreme Punishment application in C.11) that we did not talk about these ideas in the course. Such questions will explain all of the economics you need to know. You do not need to study any extra economics applications. You might find Section C.11 helpful preparation.
- Appendix D (Convex Geometry) up to Theorem D.6. Specifically, upper contour sets, quasi-convexity/concavity are not examinable.
- Appendix E (Optimisation) you should understand intuitively, but you do not need to memorise the theorems.
- Appendix G (Infinite Horizon Dynamic Programming) except the proof of Lemma G.2 (which we skipped).

## Course quality and improvement

I want this course to be of the highest possible quality, and I value your suggestions for improvement. You see last year's student survey results. I have planned the following improvements for this year:

- We will aim to give all students one-on-one informal feedback on their homework each week during tutorials.
- I am adding more applications of mathematics to economics to the notes.
- I am adding more commentary to the sample solutions (especially when students ask).
- I am adding more structure to support students applying to post-graduate degrees.