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).

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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:

  1. The language of mathematics
  2. Metric spaces (the biggest topic, about 4 weeks).
  3. Calculus
  4. Convex analysis
  5. 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:

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 visiting students, assessment is by one exam and one project. Both exams will have the same format, and last three hours:

There are several "hard" learning outcomes: being able to write down a model, being able to apply theorems to understand a model, and being able to prove theorems. I would give a passing mark (50-59) to students that can answer any sub-part that demonstrates any of these learning outcomes, e.g. to a student that writes down good model. I would give a good mark (60-69) to students who can demonstrate two of the learning outcomes, e.g. writing down a 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, especially solving a problem that draws on several mathematical ideas.

Examinable Topics