# Advanced Mathematical Economics

## Main course materials

Please regularly check that you have the
latest version the lecture notes,
which were last updated at
12:25AM, Monday 12 of October.
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 browse last year's course materials.

## News

Please check here regularly for updates. You might need to reload the page (Control-R or F5).

- Week 5:
- I am still recovering, but not from the Coronavirus. So some videos are from last year.
- Watch Dynamic programming (2.4). You can read the whiteboard.
- Watch 2019/20 week 5 lecture from 0:26:40 to 1:10 about continuity (starting C6).
- Homework: 2.13, 2.14, C.12, C.22, C.28, C.38.
- Please do the mid-semester survey. If the link doesn't work, click "Have Your Say" in Learn.

- Week 4:
- Watch Comparative statics (finishing 2.3). You can read the whiteboard.
- Watch Closed sets (C4). You can read the whiteboard.
- Watch Open sets (C5). You can read the whiteboard.
- Homework: 2.11, C.17, C.19, C.21, C.25, C.27.
- Emily Roff is running another "getting started when you are stumped" session in the The Castle (fresh). It will run for about half an hour at the start.
- The in-person tutorial today is
*cancelled*, due to illness. Please attend the online tutorial.

- Week 3:
- Watch The firm's problem (2.2). Note: this lecture includes lots of non-examinable material about the chain rule and the implicit function theorem. You can read the whiteboard.
- Watch The envelope theorem (2.3). You can read the whiteboard.
- Watch Boundaries (C3). You can read the whiteboard.
- Homework: C7, C10, C11, 2.6, 2.7, 2.9. You might want to use the Microsoft Lens app for scanning your homework.
- At the start of this week's tutorial, you are welcome to join Emily Roff in the The Castle (fresh). She will discuss last week's homework, and talk about getting started when you are stumped.
- You can sign up for the in-person tutorial. Please bring plenty of paper so you can write in a large font.
- Errata:
- In the last step of the chain rule proof of the envelope theorem (minute 26), the partial derivative should be with respect to a, not b.
- In the lazy decision maker proof of the envelope theorem, I sometimes confused Rupert Murdoch's sons, Lachlan and James. Only James is relevant to the proof.

- Week 2:
- Watch the Week 2 introduction.
- Watch Convergence (C2). You can read the whiteboard.
- Watch Production functions 1 (2.1). You can read the whiteboard.
- Watch Concave production functions (D and 2.1). You can read the whiteboard.
- Tutorials are compulsory (except they are optional for mathmicro1 students). Please read the Tutorial Guide. If you are coming to the online tutorial, please join the tutorial introduction meeting early, before it starts at 4:10pm on Thursday. After that, you will join a group. If you get "lost", you can click here to join the online tutorial.
- Homework: read Appendices E1 and E6, and do questions E1 and C5.
- The tutorial guide was just updated to give instructions on using Top Hat to confirm your attendance.

- Week 1:
- Watch AME Welcome.
- Watch Naive Set Theory (B1, B2, B3). You can read the whiteboard.
- Watch Naive Set Theory (B4, B5, B6). You can read the whiteboard.
- Watch Metric Spaces (C1). You can read the whiteboard.
- I recommend you watch the logic videos from the preparation guide.
- Homework (optional): Questions B1-14, C1, C2.
- There is no tutorial this week, but please check back regularly for details.
- Please sign up for Piazza.
- Errata:
- Part (i) of the definition of Metric Space should read "iff x=y", not "iff x=0".
- The sample solution to B.14 was wrong.
*It seems I didn't fix it properly, I will return to this soon.*

- August 11th: 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 the preparation guide 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

This will be updated at the end of lectures. In the mean time, you can refer to the examinable topics from last year.## Course quality and improvement

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

- Less jumping between topics, especially with the dynamic programming material.
- More easy questions to help the students get started.
- 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).