Advanced Mathematical Economics

Main course materials

Please regularly check that you have the latest version my lecture notes, which were last updated at 4:38PM, Thursday 06 of December. 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

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 bit about logic. 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. Time Preference   See: Debreu (1960), V19, SL4, SL5, MWG20, KK2.5
    3. Utility Maximization   See: V7, MWG3, K2.2
    4. Consumer’s Value and Policy Functions   See: V7, MWG3, K2.2, K2.3
    5. Expenditure Function and Policy Functions   See: V7, MWG3, K2.2, K2.3
    6. Slutsky Decomposition   See: V8, MWG3, K2.3
  4. 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

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:

There are three learning outcomes: ability in (1) writing models, (2) applying theorems to understand models, and (3) proving theorems. Part A is primarily about (1) and (2), and Part B is primarily about (3). I would give a passing mark (50-59) to students that can answer any question that demonstrates any of these learning outcomes, e.g. to a student that writes down a satisfactory model. I would give a good mark (60-69) to students who can demonstrate two of the learning outcomes, e.g. writing down a satisfactory model and proving that the interior and boundary of a set are disjoint. I would give a distinction (70+) to students who demonstrate all three outcomes. Getting a very high mark (90+) involves showing good knowledge in many of the mathematical topics, e.g. by solving a problem like 24.B.vii that draws on several mathematical ideas.

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

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: