Advanced Mathematical Economics
Main course materials
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- August 11th: If you are interested in taking this course, I recommend you read the preparation guide.
- I recommend you watch the logic videos from the preparation guide.
- You might consider taking the GRE test soon.
- Please sign up for Piazza.
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.
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.
- 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
- 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
- Time Preference See: Debreu (1960), V19, SL4, SL5, MWG20, KK2.5
- 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
ProjectThere is a project which is:
- compulsory for all CPD students (i.e. students enrolled in ECNM11072), and
- optional for all one-semester visiting students.
- 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 improvementI want this course to be of the highest possible quality, and I value your suggestions for improvement. You can read the survey from last year I have planned the following improvements for this year:
- I plan to take advantage of the pre-recorded videos from last year to increase the amount of contact. I plan to use extra time to do Q&A, and to do practice problems at a slow pace.
- I plan to experiment with new ways of giving homework feedback during tutorials.