# Advanced Mathematical Economics

## Main course materials

Please regularly check that you have the
latest version the lecture notes,
which were last updated at
12:10AM, Monday 15 of April.
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.

## News

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

You can browse last year's course materials, which follow a similar schedule.

- 7th of December: if you liked any of the teaching staff, please nominate them for a teaching award. This could help them get job or promotion in future.
- Week 10:
- We will cover the following topics in the lecture:
- Extreme value theorem (C9).
- Cantor intersection theorem (C.9).
- Application: Extreme punishments (C.11).
- Application: Market for lemons (C.12), time permitting.
- Note: you do not need to learn these applications. But they are good practice for applying mathematics to economics.

- Please complete the course survey.
- Homework: C.73, C.75, C.44, C.84, Practice Question 43 Part A (i).
- Quiz

- We will cover the following topics in the lecture:
- Week 9:
- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Watch Compactness introduction (C9). You can read the whiteboard.
- Watch Bolzano-Weierstrass theorem (C9). You can read the whiteboard.
- Errata:
- In the first theorem in the Compactness introduction video, there is a mistake in the proof. The fact that (X, d) is compact implies that there is a subsequence whose limit y* lies in X (not K).

- Homework: C.65, C.66, C.67, C.68, practice question 45 A(i) and 39 B(vii).
- Quiz

- Week 8:
- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Finite horizon dynamic programming (4.2).
- Watch Infinite horizon dynamic programming (4.3). You can read the whiteboard.

- Errata:
- I wrote N
_{r}(x) instead of B_{r}(x) when referring to an open ball of radius r. This is also common notation (N stands for "Neighbourhood").

- I wrote N
- Homework: 4.2, C.50, 4.4, Practice Question 39 B(viii) and 37 A(i).
- Quiz

- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Week 7:
- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Watch Complete spaces (C.7). You can read the whiteboard. Please note that understanding the proofs in this video are quite difficult. They are not essential for doing well in this course.
- Watch Banach's fixed point theorem (C.8). You can read the whiteboard. Please note that the proofs in this video are very insightful, and are very helpful for students learning how to write proofs.
- Errata:
- In the complete spaces lecture, in the proof of continuity, M and N should be the same letter.

- Homework: C.46, C.48, C.45, C.57, C.58, C.61, Practice question 6 part (i).
- If you ever have difficulty connecting to the online Friday tutorial, please email Kezhen at kchen2@ed.ac.uk
- Quiz

- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Week 6:
- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Watch Model formulation. You can read the whiteboard.
- Watch Continuity and open sets (C6). You can read the whiteboard.
- Watch Connected sets (C10). You can read the whiteboard.

- Homework: practice question 5(i), C.34, C.36, C.43, C.81, C.82.
- Quiz

- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Week 5:
- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Watch Dynamic programming (2.4). You can read the whiteboard.
- Watch Continuity (C6). You can read the whiteboard.

- Homework: Practice question 2 parts (i) and (ii), C.17, C.23, C.30, C.38, C.40.
- Errata:
- The video incorrectly states that the firm does not overshoot the production target if the production function has free disposal. This is condition is wrong. It should be that the production function is continuous.

- Quiz

- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Week 4:
- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Watch Comparative statics (finishing 2.3). You can read the whiteboard.
- Watch Boundaries (C3). You can read the whiteboard.
- Watch Closed sets (C4). You can read the whiteboard.
- Watch Open sets (C5). You can read the whiteboard.

- Homework: C.16, C.20, C.21, C.26, C.27, C.32, 2.11.
- Quiz

- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Week 3:
- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- 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 Cauchy sequences (C2). You can read the whiteboard.
- Watch Getting Started - Weeks 3-4 (optional). Note: this is helpful for the week 3 homework.
- 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.

- You can submit your homework on Learn, or on paper at the start of the lecture.
- Homework: C7, C10, C11, 2.6, 2.7, 2.9.
- Quiz
*Tutorials have moved to new rooms. See the timetable for details.*

- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Week 2:
- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Watch Convergence (C2). You can read the whiteboard. Note: I have moved some material (about Cauchy sequences) out of the Complete Spaces section into this section. I will cover these next week.
- Watch Production functions 1 (2.1). You can read the whiteboard.
- Watch Concave production functions (D and 2.1), but not quasi-convexity and upper/lower contour sets. You can read the whiteboard.
- Watch Getting started - Week 2 (optional).

- Tutorials begin this week. Tutorials are compulsory (except they are optional for mathmicro1 students). Please read the Tutorial Guide.
- Homework: read Appendices E1 and E6, and do questions E1 and C5.
- Errata:
- In the Convergence (C2) video, I wrote k instead of k² in the function convergence example.

- Quiz

- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- Week 1:
- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:
- 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.

- Watch Getting started - introduction (optional).
- Watch Getting started - week one (optional).
- I recommend you watch the logic videos from the preparation guide.
- Homework (optional): Questions B1-14, C1, C2 from the lecture notes.
- There is no tutorial this week, but please check back regularly for details.
- Please sign up for Piazza.

- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped:

## 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 (due April, 2024), and
- optional for all one-semester visiting students (due at noon on 4th of January, 2024).

## 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.
- Chapter 4 (Time), except preference relations and Debreu's theorem.
- Of Chapter 5 (Equilibrium), only model formulation is examinable (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 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, or the Market for Lemons in C.12) 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 Sections C.11 and C.12 helpful preparation.
- Appendix D (Convex Geometry) up to Theorem D.6, and also Definition D.7 and Theorem D.9. 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

This year, we are planning a few improvements:

- We will add a short quiz (not for credit) at the start of each tutorial. This will help you clear up common points of confusion quickly.
- We will pilot having an undergraduate tutor who took the course last year. We hope this will help us become more responsive to students' needs.

In the last few years, student survey response rates have been very low, so possible selection bias has prevented us from inferring students' priorities for improvement. Nevertheless, we still look for ideas in student comments. It is helpful to make the feedback very specific, e.g. instead of "some of the pictures could have been presented more clearly", it would be better to write "The picture in the proof the CB(X) is complete in the notes is unclear, but the version from the video is much clearer." We realise it isn't always possible to be specific, but it's much easier to act on specific suggestions.

- We will cover the following topics in the lecture at a leisurely
pace. If we run out out of time, please watch the videos for the
bits we skipped: