# 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, Tuesday 19 of May.
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 browse last year's course materials.

## News

- January 19th, 2018: Sample solutions and generic feedback for the exam is now available in the sample solution file above. The exam mark quintiles are 44, 58, 64, 69, 74, 83. Also, more details on how marks are calculated are available in the start of the same file. You can view your exam scripts during my office hours; please bring your student card.
- The visiting student project deadline has been extended (see below).
- December 6th, 2017: I listed the examinable topics below.
- You can watch the lecture videos on Learn by navigating to the course page and selecting "Media Hopper" on the left side.
- Week 10:
tutorial whiteboard notes
lecture whiteboard notes
- We finished Walras' law, and did 4.6 and 4.7.
- Visiting students: please send me project drafts for comments when you are ready. The final version is due on December 20th (see below).

- Week 9:
topology lecture whiteboard notes
equilibrium lecture whiteboard notes
- We finished C (topology) and started Walras' law (from 4.4).
- Homework: Practice question 12, parts (i), (ii), (iv), C.51, C.52, C.53, C.54.

- Week 8: tutorial whiteboard notes
lecture whiteboard notes
- We started C.9 and finished G.
- Homework: Practice question 5, parts (i)-(v), G.2, C.44, C.45, C.49, C.50.
- Note: I incorporated most of the tutorial whiteboard notes into the lecture notes, so I only uploaded things that are different.

- Week 7: tutorial whiteboard notes
lecture whiteboard notes
- We revisited the completeness of B(X,Y), finished C.8 and 2.4, and started G.
- Homework: 2.13, G.4, C.39, C.40, C.41.

- Week 6: tutorial whiteboard notes
topology lecture whiteboard notes
equilibrium lecture whiteboard notes
- We did C.7, 2.3, and started C.8 and 2.4.
- Homework: 2.10, C.34, C.35, C.37, C.61.

- Week 5: tutorial whiteboard notes
topology lecture whiteboard notes
equilibrium lecture whiteboard notes
- We finished C.6 and progressed a bit through 2.3.
- Homework: 2.8, C.28, C.29, C.30, C.33.
*Please check you have the latest version of the notes. The question numbers have changed.*

- Week 4: tutorial whiteboard notes
topology lecture whiteboard notes
equilibrium lecture whiteboard notes
- We did C.5, 2.2, and started C.6 and 2.3.
- Homework: Questions 2.6, 2.7, C.20, C.21, C.23, C.27. Please indicate if I can use your answers for tutorial feedback.

- Week 3: tutorial whiteboard notes
topology lecture whiteboard notes
equilibrium lecture whiteboard notes
- The lectures covered C.2, C.3, C.4, and parts of 2.1 and D.
- Homework: Questions C.7, C.9, C.10, C.13, C.15, and C.17. Please indicate if I can use your answers for tutorial feedback.

- Week 2: tutorial whiteboard notes
lecture whiteboard notes
- We are moving to bigger rooms:
- Tutorials on Thursdays: Lister Learning and Teaching Centre, Room 3.2.
- Lectures on Fridays: Chrystal MacMillan Building, Seminar Room 4.

- When you submit your homework, please tell me if you are happy (or not) with me using your work to give feedback to everyone in the class.
- We covered 4.2, 4.3, 4.5, B.4, B.5, and C.1.
- Homework: Questions C.2 and C.3. Please submit using Learn (see below). Since it's only two questions, I plan to spend about 40 minutes of the next tutorial lecturing on sections C.2 and C.3.

- We are moving to bigger rooms:
- Week 1: lecture whiteboard notes
- We covered sections B.1, B.2, B.3, B.4 and 4.1, and started 4.2.
- Optional (this week only). Homework, due Wednesday 27th of September at 11pm: prove the theorems stated in section E.2 (Optimization Transformations). Please include your student number in the subject line of the email.
- I incorrectly said that 3 out of 9 students earned firsts. It was actually 4 out of 8; the mean and range were 70.5 and 58-90, respectively. High marks are relatively easy to obtain in this course, because there are many opportunities to show you have mastered difficult ideas. For example, if you answered question 24.B.vii as well as the sample solution under exam conditions, this alone would show excellent knowledge of the third learning outcome (writing proofs).

## 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.

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. The only mathematical prerequisite for this class is high school mathematics (A-level, higher, or equivalent); if you are rusty, you might want to refresh your high school mathematics. 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",
- 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 project due at 3pm on Friday 22nd of 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. (Theorem 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).
- 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

You can send feedback directly to me or via the class representative, Lisa Wang <lisawang855@gmail.com>.

I want this course to be of the highest possible quality, and I value your suggestions for improvement. You can see this year's student survey results along with my reply. Based on the student survey from last year, I implemented the following improvements this year:

- longer tutorials,
- compulsory homework (worth 10%),
- topology spread out over the whole semester,
- mark based on the best (not average) exam,
- now available to third year students,
- more links to economics papers (time permitting).

- All topology questions now have sample solutions.
- Tutorials will have breaks after each question to accommodate:
- letting new ideas sink in,
- individual feedback,
- discussion with people near you, and
- solving new problems in the tutorial.

- Next year, I will try to get assistance from my PhD students in tutorials for more individual feedback.