# Advanced Mathematical Economics

## Main course materials

Please regularly check that you have the
latest version the lecture notes,
which were last updated at
4:40PM, Tuesday 23 of November.
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).

- 1st of June, 2021: Sample solutions are now available.
- 19th of March, 2021: In Part B, I will specify the difficulty of each question (easy, medium, hard), which specifies how much cleverness is needed to solve the question. But you might find a clever way to solve a straightforward question, so this is just a guideline.
- 10th of January, 2021: You can watch the live Q&A session.
- 15th of January, 2021: Sample solutions with commentary are now available for the December exam.
- 16th of December, 2020: Please answer the survey! It has been re-opened, and will close again on the 3rd of January in the evening.
- Week 10:
*Everyone,*please sign-up for a tutorial, either morning (online), afternoon (online), or afternoon (in-person).- Homework: Practice question 31, C.63, C.70, C.74, C.75.
- Bonus lecture, not examinable: Connected sets (C10). You can read the whiteboard.
- Bonus lecture, not examinable: 2018/9 lecture 10 from 0:38:30 to 1:03:00 about the Heine-Borel theorem (C9). You can read the whiteboard.
- The live question and answer lecture on Tuesday didn't record
properly.
But you can read the
whiteboard here.
Here's a brief summary:
- We went through the revised sample solution to question 4.4.
- We talked about the philosophy behind the d
_{1}and d_{∞}metrics. - We talked about study resources: old exams (which all have sample solutions), Piazza, office hours, other students.
- We talked about applying to post-graduate programmes, and recommendation letters. Dissertations matter a lot if you are applying directly to PhD programmes. Letters are very important, because of grade inflation, and because "mathematical economics" can mean a lot of things. I can write letters for everyone!

- I have published the list of examinable topics below.
- Please complete the Course Enhancement Questionnaire by logging into Learn, and clicking "Advanced Mathematical Economics", "Have Your Say", and "Course Enhancement Questionnaires (CEQs)".

- Week 9:
- Next week (week 10), we will have a live question-and-answer session starting at 4pm on Tuesday.
*Everyone,*please sign-up for a tutorial, either morning (online), afternoon (online), or afternoon (in-person). Please note that since more people are coming to the online afternoon tutorials, Emily Roff will move back to these.- Watch Compactness introduction (C9). You can read the whiteboard.
- Watch Bolzano-Weierstrass theorem (C9). You can read the whiteboard.
- Watch 2019/20 week 10 lecture from 0:31:05 to 0:43:05 about the Extreme Value Theorem (C9), and from 1:10:15 to 1:30:30 about extreme punishments (C11). You can read the whiteboard.
- Homework: part A of practice question 20, C.50, C.54, C.59, C.62, C.66 (skip home production).

- Week 8:
- Watch 2019/20 week 8 lecture from 0:56:20 about infinite horizon dynamic programming (continuing 4.3) and Banach's fixed point theorem (C.8). You can read the whiteboard.
- Watch 2019/20 week 9 lecture from 0:14:20 until 0:46:20 about infinite horizon dynamic programming (finishing 4.3). You can read the whiteboard.
- Watch Model formulation (loosely based on 5.3). You can read the whiteboard.
*Everyone,*please sign-up for a tutorial, either morning (online), afternoon (online), or afternoon (in-person).- Homework: part (i) of practice question 2, and C.37, C.42, C.46, C.56, 4.4.

- Week 7:
- Watch 2019/20 week 6 lecture from 1:15 to 1:28 about completeness (continuing C7). You can read the whiteboard.
- Watch 2019/20 week 7 lecture until 0:58 about completeness (finishing C7). You can read the whiteboard.
- Watch 2019/20 week 8 lecture from 0:43:15 to 0:50:15 about infinite horizon dynamic programming (starting 4.3). You can read the whiteboard.
- Homework: C.27, C.31, C.38, C.41, C.44, C.45
- Please
sign-up for a tutorial, either morning (online), afternoon
(online), or afternoon (in-person).
*Update: I had limited the number of people in the online tutorials to 3 by mistake. Please try again.* - This week, Reuben Wheeler will give a short talk in the morning and in the afternoon at the start of the online tutorials. He will talk about how you can study in groups online.

- Week 6:
- Watch 2018/19 week 7 lecture from 1:25 to 1:47 about finite horizon dynamic programming (4.2). You can read the whiteboard.
- Watch 2019/20 week 5 lecture from 1:10 to 1:34 about continuity (continuing C6). You can read the whiteboard.
- Watch 2019/20 week 6 lecture from 0:00:49 to 0:46 about continuity (finishing C6). You can read the whiteboard.
- Watch 2019/20 week 6 lecture from 0:51:55 to 1:15 about completeness (starting C7). You can read the whiteboard.
- Homework: 4.2, C.23, C.30, C.32, C.36, C.37 (indifference curves only).
- In-person tutorial: please sign up here if you want to come.
- Following the mid-semester survey:
- There will be an extra online tutorial at 10am. Chris Stapenhurst will be available to answer economics questions in the second hour of both online tutorials.
- In-person tutorials are below capacity, so you can come every week.
- We will ask you to sign up for one of the three options. Please consult your group, or switch groups.
- Please email Chris Stapenhurst if you need help finding a suitable group.

- 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). You can read the whiteboard.
- Homework: 2.13, 2.14 parts (i) and (iii), 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* - Utility Maximization
*See: V7, MWG3, K2.2* - Consumer’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
- Time
- Time Preference
*See: Debreu (1960), V19, SL4, SL5, MWG20, KK2.5*

- Time Preference
- 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

## Project

There is a project which is:- compulsory for all CPD students (i.e. students enrolled in ECNM11072), and
- optional for all one-semester visiting students.

## Examinable Topics

- Chapter 2 (Production) except the following:
- Quasi-concavity and upper contour sets.
- The constrained envelope theorem, i.e. none of Section 2.5.
- Production technology sets, i.e. none of Section 2.6.

- None of Chapter 3 (Consumption) is examinable.
- All of Chapter 4 (Time) is examinable.
- None of Chapter 5 (Equilibrium) is examinable, except model formulation (which isn't really explained in the notes anyway).
- 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.

## 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 and my response to this year's mid-semester survey. 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).