Advanced Mathematical Economics

Main course materials

Please regularly check that you have the latest version the lecture notes, which were last updated at 10:33PM, Tuesday 18 of June. 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.

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


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: Two of these books -- Daepp and Gorkin (2011), and Kane (2016) -- have substantial sections on how to write proofs in the context of metric spaces, so these might be a good match.

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.

  1. Introduction
  2. Production
    1. Production Functions   See: V1, MWG5, K7.1
    2. Profit Maximization   See: V2, MWG5, K7.2
    3. Upper Envelopes and Value Functions   See: V3, SL4, MWG5, K7.2
    4. Cost Functions and Dynamic Programming   See: V4, SL4, MWG5, K7.3, K.A.2
    5. Upper Envelopes with Constraints   See: V5, SL4, MWG5, K7.3
  3. Consumption
    1. Utility Functions   See: V7, MWG3, K2.1
    2. Utility Maximization   See: V7, MWG3, K2.2
    3. Consumer’s Value and Policy Functions   See: V7, MWG3, K2.2, K2.3
    4. Expenditure Function and Policy Functions   See: V7, MWG3, K2.2, K2.3
    5. Slutsky Decomposition   See: V8, MWG3, K2.3
  4. Time
    1. Time Preference   See: Debreu (1960), V19, SL4, SL5, MWG20, KK2.5
  5. Equilibrium
    1. Economies   See V17, V18, V19, MWG16, K6.1, K6.2
    2. Efficient Allocations   See V17, V18, V19, MWG16, MWG22, K5.2
    3. Equilibrium   See V17, V18, V19, MWG16, K6.1
    4. Characterising Equilibria   See V17, MWG15, MWG16, MWG20, K2.2
    5. Efficiency of Equilibria   See V17, MWG16, K6.3
    6. *Existence of Equilibria   See V17, MWG17, K6.4
    7. Implementation of Efficient Allocations   See V17, MWG16, K6.3


There is a project which is: You are encouraged to send me drafts for comments.

Examinable Topics

Course quality and improvement

Only 13 students answered the mid-semester survey. Most students had very positive things to say about the course. The main suggestions for improvement were:

The following changes are planned for this year: