Syllabus

Introduction

Here is a syllabus for a five-week, full-time mathematical economics course, at Masters level. This course should be held before the semester begins, so that students are equipped with the necessary mathematical tools to approach the quantitative components of those future courses. For example, Macroeconomics requires Optimal Control Theory (use of Hamiltonians, the dynamic optimization, which is the more advanced concept compared to Lagrangians, the static optimization). Another example, econometrics requires aptitude in Linear Algebra. 

Required Textbooks

In the image to the right, the most essential textbooks are on the left (e.g. Linear Algebra, Introductory Statistics, Calculus and a bit of Set Theory). In the middle, the more tricky part of this course, you find Real Analysis. On the right-hand side of the image, I just included some extension textbooks, which aren't part of the course. Metric Spaces and Abstract Algebra.

You do not need to read all the chapters below. This just provides a guide about what is in our syllabus. I highly recommend that you read this section of the course outline (Required Textbooks), so that you know what is in the syllabus. I also recommend that you scan the textbooks, to figure out which material you are comfortable with, and which material you need to spend more time on. This outline provides a comprehensive and supportive outline of what is in the syllabus. I have read all the material in this outline. I have selected the textbooks that I used, when learning the topics. In the maths major in the Mathematics Department at UCT, Serge Lang's Real Analysis textbook was used by A/Prof Alexandar Ianovsky, in 2015. The Anton & Rorres (2011) Linear Algebra textbook was recommended by Stellenbosch University in the mathematical sciences major (Commerce Faculty), in first year, 2013. Even though Dr Jesse Ratzkin, in second year maths major, 2015, didn't use the Anton & Rorres (2011) textbook, I still found that textbook very helpful, giving a clear description of Linear Algebra. Dr Ratzkin used his own notes.

Visit Library Genesis http://libgen.is . Please purchase physical textbooks when you think that you will end up reading the whole book. There's not much point purchasing the ebook (due to the URL at the beginning of this paragraph), unless the ebook is not in Library Genesis. I once bought an economics ebook (I'm not going to say which one), for this reason, as I like reading PDFs (obviously, with dark mode - you should make the background on the PDF black, and the text completely white, for example with Xodo Reader (download the Windows app, or the mobile device app)). If you have a tablet, you could use Xodo to read the textbooks, at your leisure, sitting on the couch in the evening with a glass of Amarula. Or, you could also use your tablet to browse newspapers on the PressReader app, after connecting to UCT's VPN. But I understand that not everybody has a tablet (perhaps too many people think that a tablet is for games).

Econometrics (ECO5046F and ECO5070S)

Ben Lambert: A Graduate Course in Econometrics (YouTube playlist). (Also Graduate Econometrics on the Ox educ channel.)

Examples of physical textbooks that you should order through Amazon.com, find in a local second-hand dealer, or at Protea Books, include Wooldridge's "Baby Econometrics" Textbook (Honours), and "Big Wooldridge" (Econometric Analysis of Cross Section and Panel Data. It's heavy.). But those textbooks are for econometrics only, so they come after the ECO5011F course.

Linear Algebra

3Blue1Brown: Essence of Linear Algebra (YouTube playlist).

Anton, H. & Rorres, C. 2011. Elementary linear algebra: with supplemental applications. (10th ed.). S.l.: John Wiley & Sons (Asia) Pte Ltd.

One copy available in the main library, 512.5 ANTO. Newer versions available in the main library, same Dewey Decimal, including an electronic version.

Prerequisites (from Honours Quants course)

Recommended

Linear independence; basis; dimension; row space, column space and null space; rank, nullity and the fundamental matrix spaces; matrix transformations from n dimensions to m dimensions; geometry of matrix operators; dynamical systems and Markov Chains.Dynamical systems and Markov Chains is important for economic modelling. Researchers who have a PhD would be able to use Leontief Input-Output models (akin to Social Accounting Matricies), to set up a computable general equilibrium (CGE) model.
Orthogonality is part of Ordinary Least Squares (OLS).
Again, orthogonality is part of Ordinary Least Squares (OLS).
Part of the second semester microeconometrics course, when theoretical methods to find standard errors is discussed, for example, the Wald Statistic.

Prerequisites (1st year Statistics)

Underhill, L. & Bradfield, D. 2014. IntroStat. Department of Statistics, University of Cape Town.

Compulsory (lectures)

Wittenberg, M. 2011. Econometric Theory. School of Economics, University of Cape Town. Chapters 1–5.

This literature is not in the public domain—it is only in the UCT Economics domain.

Aidan Horn's lecture notes for the course (from 2019) including Edwin's section.

The rest of the chapters (after chapter 5) will be used for ECO5046F (Advanced Econometrics).

Lab Sessions & Homework (entire semester—ECO5011F + ECO5046F)

Wittenberg, M. 2010. Econometrics through applications: A practical handbook. School of Economics, University of Cape Town.

Mr Aidan Horn provides solutions (plug). Last revised: Semester 1, 2019.


Prerequisite: Set Theory

Hrbacek, K. & Jech, T. 1999. Introduction to set theory: third edition, revised and expanded. New York: Marcel Dekker, Inc.

UCT Library: digital version via Elsevier ScienceDirect (or acquire via Library Genesis).

Descriptive interlude: Logic 101

The following logical terminology is important for this course.

If a premise A implies a conclusion B, then that is the same thing as writing A ⇒ B, or the same thing as saying, "If A then B." The premise A is then sufficient for the conclusion to be true. Also, the conclusion B is necessary for the statement A ⇒ B to be true, and we can say, "B only if A". 

If an event X is both necessary and sufficient for Y, then we can write X ⇔ Y. In words, "X if and only if Y", or shorthand, "X iff Y". This concept is called material equivalence of X and Y.

Set Theory uses other symbols, such as ¬ (negation of logical statement, such as "¬A"); \ (not, with sets, for example "A\B"); ∧ (and, with logical statements); ∨ (or, with logical statements); ∩ (intersection of sets); ∪ (union of sets); ∅ (the empty set); etc. For more symbols, see the Mathematical Symbols chapter at the end of Chiang and Wainwright (2005).

Compulsory: Real Analysis

Lang, S. 2005. Undergraduate analysis: second edition. New York: Springer.

Three copies available on short loan at the UCT Chancellor Oppenheimer Library (COL), at 515.8 LANG.

Part One: Review of Calculus

Recommended

Part Two: Convergence

Compulsory

Part Four: Calculus in Vector Spaces

Supplementary: Calculus

Stewart, J. 2010. Calculus: concepts and contexts. (4th Metric International Edition). Belmont: Brooks/Cole.

This was the prescribed textbook for MAM1000W and MAM2000W. Available widely from second-hand book stores, and from 515 STEW in the COL (three available on long loan; four in the short loans department). Please ask underground or on Library Genesis for the PDF. 

Prerequisites (1st year mathematics)

Compulsory

Dynamic Optimization

Hamiltonians are optimal time paths of a production problem. In comparison, the method of Lagrangian multipliers (from second and third year economics) optimizes a static point in time. However, dynamic optimization optimizes the time paths of the control, state and costate (shadow price) variables, over time.

Prerequisites (Honours Quants course)

Chiang, A.C. & Wainwright, K. 2005. Fundamental methods of mathematical economics. (4th Ed.). Boston, MA: McGraw-Hill Education.

Five available on short loan at the main library, at 330.0151 CHIA. One available on long loan. A solution manual also exists.

Part One: Introduction

Part Two: Static (or Equilibrium) Analysis

Method for finding a steady-state Markov transition matrix.
5.1: Nonsingularity of matricies (i.e. invertible); 5.2: Determinants aren't used by practicing economists, because the statistical computer program has that coded in by the statisticians who write the commands.; 5.3: Essential knowledge about how solutions to linear systems are found.; 5.4: An important method to learn about how to invert a matrix, via the conjugate transpose. But, please note that inversion via Gauss reduction in an augmented coefficient matrix (starting with an identity matrix to the right of the | equals sign) is quicker and simpler! E.g. for small 3x3 matricies.; 5.5: Cramer's rule gives a shortcut somewhat, but again, you won't use this piece of mathematical economics in practice.; 5.6: Foundational linear algebra modelling, essential for macroeconomics.; 5.7: Necessary for social accounting matricies (SAMs). See chapter 4 and §10.5 from Anton and Rorres (2011), mentioned above.; 5.8: A short paragraph on the theoretical limitations of static analysis, in a dynamic world.

Co-requisites

Part Three: Comparative-Static Analysis

Compulsory

Part Four: Optimization Problems

13.1: Nonlinear Programming and Kuhn-Tucker Conditions; 13.2: The Constraint Qualification; 13.3: Economic Applications; 13.4: Sufficiency Theorems in Nonlinear Programming;

Extension

13.5: Maximum-Value Functions and the Envelope Theorem (although Interpretation of the Lagrangian Multiplier is compulsory knowledge).

Compulsory

Part Five: Dynamic Analysis

Extension (for macroeconomists; not in the syllabus)

Compulsory


Chiang's second mathematical economics textbook (where stuff gets tricky)

This part of the course is not taught in undergraduate mathematics; thus, it is post-graduate mathematics. These chapters are very tricky, but they are summarized in Edwin's presentation slides, to make it easier for you. So, his slides should guide you with the essentials, and what parts of the content to focus on. I haven't even read this textbook (because it is long; I've skim-read it), although I will probably end up reading it some day because I want to teach this course in future. That being said, this textbook is part of the syllabus for this course, and everything outlined below is very important for the content in this course—you'll probably just find it quicker to pick up from Edwin's slides. The key idea here is not to parrot-learn, but to understand this topic.

Chiang, A.C. 1992. Elements of Dynamic Optimization. Singapore: McGraw-Hill, Inc.

One copy available in the main library, 330.0151 CHIA. Amazon.com: 157766096X.

Compulsory

Part 1: Introduction

1.1: Salient Features of Dynamic Optimization Problems; 1.2: Variable Endpoints and Transversality Conditions; 1.3: The Objective Functional; 1.4: Alternative Approaches to Dynamic Optimization.

Part 2: The Calculus of Variations

2.1: The Euler Equation; 2.2: Some Special Cases; 2.3: Two Generalizations of the Euler Equation; 2.4: Dynamic Optimization of a Monopolist; Extension only: 2.5: Trading Off Inflation and Unemployment
You might just end up trying to memorize rules-of-thumb here, instead of trying to understand the logic behind the transversality conditions (I personally also haven't grasped this chapter yet, myself, but I will try to learn it and revise it. Watch this space.).3.1: The General Transversality Condition; 3.2: Specialized Transversality Conditions; 3.3: Three Generalizations; 3.4: The Optimal Adjustment of Labor Demand.
5.1: Methodological Issues of Infinite Horizon; 5.2: The Optimal Investment Path of a Firm; 5.3: The Optimal Social Saving Behavior; 5.4: Phase-Diagram Analysis (see supplementary text by Strogatz below, which will help you with phase diagram analysis); 5.5: The Concavity/Convexity Sufficient Condition Again.
6.1: Four Basic Types of Constraints; 6.2: Some Economic Applications Reformulated; 6.3: The Economics of Exhaustible Resources.

Part 3: Optimal Control Theory

7.1: The Simplest Problem of Optimal Control; 7.2: The Maximum Principle; 7.3: The Rationale of the Maximum Principle; 7.4: Alternative Terminal Conditions; 7.5: The Calculus of Variations and Optimal Control Theory Compared; 7.6: The Political Business Cycle; 7.7: Energy Use and Environmental Quality.
8.1: An Economic Interpretation of the Maximum Principle (including The Costate Variable as a Shadow Price); 8.2: The Current-Value Hamiltonian; 8.3: Sufficient Conditions; 8.4: Problems with Several State and Control Variables; 8.5: Antipollution Policy.

The following chapters can be defined as the "curve-ball" exam questions, the "last 10%". I.e. the questions for the boffins. Not that I would be able to solve them first time myself! There might be curve-balls in the exam, but please don't be put-off too much if you can't solve these questions. You could always come back to this course later in your career, if you need to know more about dynamic optimization, for the work that you will do in future.

9.1: Transversality Conditions (in an infinite horizon context); 9.2: Some Counterexamples Reexamined (including the Halkin counterexample); 9.3: The Neoclassical Theory of Optimal Growth; 9.4: Exogenous and Endogenous Technological Progress.
10.1: Constraints Involving Control Variables; Solving Halkin problems or Hamiltonians requires the use of linear algebra, as one can Gauss-reduce the system of differential equations.10.2: The Dynamics of a Revenue-Maximizing Firm; 10.3: State-Space Constraints; 10.4: Economic Examples of State-Space Constraints; 10.5: Limitations of Dynamic Optimization.

Finito!

Phase diagrams (compulsory)

Strogatz, S.H. 1994. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering. Reading, MA: Perseus Books Publishing.