## Advanced Mathematical Economics

## Course materials

Please regularly check that you have the latest version my lecture notes, which were last updated at 11:16AM, Friday 15 of November.

You can see what changed using Adobe Acrobat Pro on the uCreate computers.
Choose `Tools -> Compare Documents`

, and select the PDFs containing
the old and new versions of the notes.

You can download the practice questions and solutions, including the mock exam (question 20).

## News

- 17th of January, 2017 at 10am: I will have a special office hour where you will be able to look through your exams, and I will talk through the sample solutions. (You can come to any office hour to see your exam, but you will learn more if you come to this one.)
- 10th of January, 2017: exam results are now available on learn, and sample solutions to the exam questions are now available in the practice question files above. Students are welcome to come to my office hours until the final exam in May (and beyond). I will also organise a time to go over the exam questions.
- Week 10:
- I have uploaded a mock exam, with sample solutions (see above).
- You can download the week 10 tutorial sample solutions and watch the week 10 tutorial.
- In lecture, we carefully defined derivatives, and revisited the envelope theorem, this time studying when value functions are differentiable (Appendix F.1 and F.4). You can read the whiteboard notes and watch the lecture.

- Week 9:
- You can download the week 9 tutorial sample solutions and watch the week 9 tutorial video.
- In lecture, we covered infinite horizon dynamic programming (Appendix G). You can read the whiteboard notes and watch the lecture.
- Homework: Micro1 Practice Questions 1 (i-iv) and 6 (i, iii, iv, v), and Questions G.1, G.4, C.72.

- Week 8:
- You can download the week 8 tutorial sample solutions. Unfortunately, the video recording did not work.
- In lecture, we covered dynamic programming in the contexts of the profit maximisation problem (section 2.4) and the cake-eating problem (section 3.2). We also proved that the marginal cost curve is increasing (section 2.5). You can read the whiteboard notes and watch the lecture.
- Homework: Questions 2.13, 2.14, 2.16, A.64, A.65, A.66.
*Please check you have the latest version of the notes.*

- Week 7:
- You can download the week 7 tutorial sample solutions and watch the week 7 tutorial video.
- We finished Section 2.3 (envelope theorem and convexity of value functions) in lecture. You can read the whiteboard notes and watch the lecture.
- Homework: Questions 2.8, 2.9, 2.10, A.59, A.60, A.62.

- Week 6:
- You can download the week 6 tutorial sample solutions and watch the week 6 tutorial video.
- In lecture, we covered Section 2.1 (production functions, excluding upper contour sets), Appendix A.3 up to Theorem A.29 (convexity), and up to Section 2.2 (stating but not proving the envelope theorem). You can read the whiteboard notes and watch the lecture (the microphone ran out of battery 1 minute in).
- Homework: Questions 2.1, 2.2, 2.7, A.11, A.15, A.27, A.58.

- Week 5:
- You can download the week 5 tutorial sample solutions and the week 2 tutorial sample solutions, which answer the homework from weeks 1 and 4 respectively.
- We covered A.2.9 (Compactness). The main theorems were: Bolzano-Weierstrass, Weierstrass, Heine-Borel, and the Cantor intersection theorem. The you can watch the first half of the lecture 5 (before the computer broke down), and read the board notes of lecture 5.
- Homework: Questions A.43, A.44, A.45, A.47, A.49, A.51. As usual, I recommend you send me your solutions by Wednesday evening.

- Week 4:
- You can download the week 4 tutorial whiteboard notes, which answer the homework from week 3.
- You can download the lecture 4 whiteboard notes and watch the lecture 4 video.
- We covered Appendices A.2.6 and A.2.7 (Cauchy sequences, complete metric spaces, Banach's fixed point theorem).
- Homework: Questions A.33, A.35, A.37, A.39, A.40, A.41.
Please download the latest version of the notes.
I invite you to email me your solutions by Wednesday evening before
the tutorial, so I can give you better feedback.
*I simplified question A.41, and fixed big mistakes in questions A.33 and A.39 so please check you have the latest version.*

- Week 3:
- You can download the week 3 tutorial whiteboard notes, which answer the homework from week 2.
- You can download the lecture 3 whiteboard notes.
- We covered Appendix A.2.5 - A.2.6 (Open sets, Continuity).
- Homework: Questions A.20, A.23, A.24, A.28, A.29, A.32. Please make sure you have the latest version of the notes.
- I added some books about writing proofs in the Extra Reference section below.
- Jakub Zowczak (s1347775) kindly offered to be the class representative. You can send feedback directly to me, or via Jakub.

- Week 2:
- You can download the lecture 2 whiteboard notes.
- We covered Appendix A.2.1 - A.2.4 (Metric Spaces, Sequences and Convergence, Boundaries, Closed sets).
- Homework: Questions A.3, A.7, A.8, A.14, A.16, A.17.
- Note: attendance at tutorials (but not lectures) is compulsory, because practice and feedback are critical for this course.

- Week 1:
- You can download the lecture 1 whiteboard notes.
- We covered Appendix A.1.1 - A.1.5 (Naive Set Theory), Definitions 4.1, 4.3, 4.4, 4.5, 4.8 (pure-exchange economy definitions), and Theorem 4.3 (First Welfare Theorem).
- For homework, please try to prove the 5 theorems in section A.4.2, (including the last question). Don't worry if you get stuck; we will spend most of the tutorial on this.

## Overview and materials

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 (when appropriate).

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 Topics in Microeconomics 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. This is the first time the course has run, so there are no past exam papers. However, I expect the assessment will overlap about 50% with my post-graduate microeconomics course, which has many past exam papers with sample solutions.

I plan to proceed through the following topics:

- The language of mathematics
- Metric spaces (the biggest topic, about 4 weeks).
- Calculus
- Convex analysis
- Dynamic programming

## Extra Reference

Some students like an extra reference, although it is unnecessary. I will only test knowledge from my notes, and I will list the examinable concepts in due course.

For the first two topics (the language of mathematics and metric spaces), the closest book to my notes is Rosenlicht's (1968) "Introduction to Analysis". For the next two topics (calculus and convex analysis), the closest book to my notes is Boyd and Vandenberghe's (2004) "Convex Optimization". For the last topic (dynamic programming), 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 visiting students, assessment is by one exam and one project. Both exams will have the same format, and last three hours:

- Part 1 would be of a similar format to one of my Microeconomics 1 exam questions. There would be an informal description of a social problem, which the student would have to translate into an economic model. The subsequent parts would 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 2 would be purely mathematical. 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.

## 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) except the following:
- The proof of the open/closed set characterisations of continuity. (Theorem C.7)
- The proof that the bounded functions, and continuous and bounded functions form a complete metric space. (Theorem C.13)
- The proof of the Bolzano-Weierstrass Theorem. (Theorem C.15)

- 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 F (Calculus) sections F.1 (Foundations) and F.4 (Envelope Theorem), except for the proof of the Benveniste-Scheinkman theorem. You should also understand how to use the chain rule, but not necessarily in terms of matrices.
- 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 continuous and bounded functions.