# Advanced Mathematical Economics

## Main course materials

Please regularly check that you have the
latest version my lecture notes,
which were last updated at
12:00AM, Thursday 26 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 practice exam questions and sample solutions with commentary.

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

- Week 10: whiteboard notes
- We did C9 and bonus material in C11.
- Homework: C59, C63, C65, C66, C71. Optional: work on practice question 29 in tutorial to get feedback.

- Week 9: whiteboard notes
- We talked about Part A of the exam (following the notes in the practice questions file), finished G and started C9.
- Homework: Practice question 12, parts (i), (ii), (iv), G1, G2, G3, G4, C57.

- Week 8: whiteboard notes
- We did C8 and progressed through G.
- Please get the latest version of the notes; I made many minor changes recently in C8 and G.
- Homework: Practice question 5 parts i-v, C49, C50, C52, C54.

- Week 7: whiteboard notes
- We did 2.3, 2.4, 3.2 (skipping everything up to Theorem 3.3), and started G.
- Homework: 2.11, 2.14, C36, C46, C48
- I corrected the sample solutions for C42 and C48.

- Week 6: whiteboard notes
- We did C7 and started 2.3.
- Homework: C38, C39, C40, C41, C42, C44.

- Week 5: whiteboard notes
- We did C6 and 2.2.
- Homework: C31, C32, C33, redo C24 using continuity, 2.6, 2.7.

- Week 4: whiteboard notes
- We did C5, 2.1 and D.
- Homework: Questions C22, C23, C25, C26, C27, C29.
- Tutorial reminder: Tutorials are for feedback. Please bring anything you would like feedback on.

- Week 3: whiteboard notes
- We did C2, C3, C4, and started 2.1.
- Homework: Questions C2, C5, C7, C11, C16, C17, C18.

- Week 2: whiteboard notes
- We did (parts of) sections B3, B6, 4.5, C1.
- Homework: Question E.1, and prove theorems E.5, E.6, E.7, E.8, E.9. Please work on these during this week's tutorial. Please submit your answers (or explanations of where you got stuck) next lecture on paper or via Learn.
- Please bring all of your homework you have done so far to the tutorial.

- Week 1: whiteboard notes
- We did parts of sections B1, B2, B4, B5, 4.1, 4.2, 4.3. The logic parts were just a quick introduction; please watch the videos.
- Homework (not for submission this week): Questions B.1 - B.14. Please ask for help on Piazza.
- There is no tutorial this week.

- August 27th: I added information about (a) Piazza and (b) Continuing Professional Development.
- August 17th: I uploaded a preparation guide for the course.

## Overview

This course teaches some of the important mathematics 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 and to Continuing Professional Development students who are not enrolled on any degree. The Continuing Professional Development programme is running as a pilot this year, so please email me if you are interested. Continuing Professional Development students can simultaneously enrol in School of Mathematics courses such as Introduction to Linear Algebra, Calculus and its Applications, Proofs and Problem Solving, Accelerated Algebra and Calculus for Direct Entry, Accelerated Proofs and Problem Solving.

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. 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

## Assessment

For the non-visiting students, all assessment will be by two exams, one in December, and one in May. For one-semester visiting students, assessment is by one exam and one optional project due in mid December; you are welcome to send me drafts for comments. Both exams will have the same format, and last three hours:

- Part A begins with translating an informal description of a social problem into an economic model. The subsequent questions involve applying theorems to learn about the model, e.g. "prove that there is net migration from the small country to the large country", or "devise a lump-sum tax regime to deliver an egalitarian equilibrium."
- Part B consists of independent questions, and is more focused on writing mathematical proofs. Some questions would be to prove things that were not covered in the class (but are based on the same ideas). Some questions would be to provide examples or counter-examples.

The weekly homework problems are compulsory, and are worth 10% of your mark. They are not assessed, I only want to see that you attempted most of the questions. Please submit your homework using Learn. You can either type or scan/photograph your homework. Please indicate on your assignment if I can use your assignment anonymously to give feedback to the whole class.

## 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)

- Most of Chapter 3 (Consumption) is not examinable. Only the cake-eating problem is examinable (Section 3.2).
- Most of Chapter 4 (Equilibrium) is not examinable.
These parts are examinable:
- Pure exchange economies and feasible allocations (Definition 4.1)
- Utility possibility set, Pareto dominance, Pareto efficient, Pareto frontier, Social Welfare Function. (Definitions 4.3-4.7)
- Pure-exchange equilibrium. (Definition 4.8)
- Efficiency of equilibria. (Section 4.5)

- 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 proof of the Heine-Borel theorem.
- 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 you do not need to remember how to prove that the Bellman operator in the cake-eating problem is a self-map on the space of continuous and bounded functions.

## Course quality and improvement

I want this course to be of the highest possible quality, and I value your suggestions for improvement. You can read my discussion of the mid-semester survey. You can also see last year's student survey results along with my reply. I am implementing the following improvements this year:

- more logic material, both in the lecture notes and in the preparation guide,
- links in each section of the lecture notes to relevant past exam questions,
- fairer assessment for visiting students, mirroring that of local students,
- scheduling lectures and tutorials at more sociable hours,
- allocating more tutors to give more individual feedback (resources permitting),
- we will use the Piazza website to facilitate more feedback, both from the lecturer and from fellow students,
- piloting enrolment for Continuing Professional Development students.