# 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 gen.lib.rus.ec (seems it redirects to http://libgen.rs/ now). 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)

Chapter 1: Systems of Linear Equations and Matrices

Chapter 2: Determinants (not necessary for Masters Economics, but is a foundation)

Chapter 3: Euclidean Vector Spaces (you will encounter this in Masters Econometrics)

### Recommended

Chapter 4: General Vector Spaces (advanced Masters Econometrics)

Chapter 5: Eigenvalues and Eigenvectors

Chapter 6: Inner Product Spaces

Chapter 7: Diagonalization and Quadratic Forms

Chapter 8: Linear Transformations

Chapter 10: Applications of Linear Algebra. Only the following sections:

10.2: Geometric Linear Programming

10.3: The Earliest Applications of Linear Algebra

10.4: yes, splines are used at PhD level.

10.5: Markov Chains. See description below Chapter 4 (above).

10.6: Graph Theory. Only useful to design the structure of the internet. For example, Google's search algorithm, or Google Maps finding the best route for you to take when you drive.

10.7: Games of Strategy. Otherwise known as Game Theory (microeconomics).

10.8: Leontief Economic Models.

Modelling the economy.10.9: Forest Management. If you're a natural resource economist.

10.14: Chaos. PhD macroeconomics.

10.17: Age-Specific Population Growth. Population growth affects development economics.

10.18: Harvesting of Animal Populations. Natural resource economics.

10.20: Warps and Morphs can be used in artificial intelligence (AI), but not to do with economics. Unless you're scanning satellite imagery to create economic indicators. In that situation, perhaps the economists would use the data, but leave the AI to computer scientists.

Appendix A: How to Read Theorems.

- Statistics

### 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).

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.

- Real Analysis, Dynamic Programming and Optimal Control Theory

### 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).

Chapter 1: Sets

Chapter 2: Relations give a foundational understanding of functions.

Chapter 3: Natural Numbers. We encounter natural numbers in real analysis.

Chapter 4: Finite, countable, and uncountable sets. The number lines that we use in economics are typically infinitely uncountable.

### 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

Chapter 0: Sets and Mappings

Chapter I: Real Numbers

Chapter II: Limits and Continuous Functions

Chapter III: Differentiation

Chapter IV: Elementary Functions

Chapter V: The Elementary Real Integral

### Recommended

Part Two: Convergence

Chapter VI: Normed Vector Spaces. This is similar to Linear Algebra, just more abstract.

Chapter VII: Limits.

### Compulsory

Chapter X: The Integral in One Variable.

Part Four: Calculus in Vector Spaces

Chapter XV: Functions on n-Space.

Chapter XVII: Derivatives in Vector Spaces.

Chapter XIX: Ordinary Differential Equations.

### 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)

Chapter 0: A Preview of Calculus.

Chapter 1: Functions and Models.

Chapter 2: Limits and Derivatives.

Chapter 3: Differentiation Rules.

### Compulsory

Chapter 4: Applications of Differentiation.

Chapter 5: Integrals.

Chapter 7: Differential Equations.

Chapter 9: Vectors and the Geometry of Space.

Chapter 10: Vector Functions.

Chapter 11: Partial Derivatives.

Including 11.8: Lagrange MultipliersChapter 12: Multiple Integrals.

Chapter 13: Vector Calculus.

Appendixes: A–F.

## 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

Chapter 1: The Nature of Mathematical Economics

Chapter 2: Economic Models

Part Two: Static (or Equilibrium) Analysis

Chapter 3: Equilibrium Analysis in Economics

Chapter 4: Linear Models and Matrix Algebra

Chapter 5: Linear Models and Matrix Algebra (Continued)

### Co-requisites

Part Three: Comparative-Static Analysis

Chapter 6: Comparative Statics and the Concept of Derivative

Chapter 7: Rules of Differentiation and their use in Comparative Statics

Chapter 8: Comparative-Static Analysis of General-Function Models

### Compulsory

Part Four: Optimization Problems

Chapter 9: Optimization: A Special Variety of Equilibrium Analysis

Chapter 10: Exponential and Logarithmic Functions

Chapter 11: The Case of More Than One Choice Variable

Chapter 12: Optimization with Equality Constraints

Chapter 13: Further Topics in Optimization

### Extension

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

Part Five: Dynamic Analysis

Chapter 14: Economic Dynamics and Integral Calculus

Chapter 15: Continuous Time: First-Order Differential Equations

### Extension (for macroeconomists; not in the syllabus)

Chapter 16: Higher-Order Differential Equations

Chapter 17: Discrete Time: First-Order Difference Equations

Chapter 18: Higher-Order Difference Equations

Chapter 19: Simultaneous Differential Equations and Difference Equations

### Compulsory

Chapter 20: Optimal Control Theory

Mathematical Symbols

Recommended Reading by Chiang and Wainwright: A Short Reading List

## 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

Chapter 1: The Nature of Dynamic Optimization

Part 2: The Calculus of Variations

Chapter 2: The Fundamental Problem of the Calculus of Variations

Chapter 3: Transversality Conditions for Variable-Endpoint Problems

Chapter 4: Second-Order Conditions

Probably not heavily weighted in the syllabus, but still in the syllabus. Very tricky! Rather skip this chapter if you are struggling.Chapter 5: Infinite Planning Horizon

Chapter 6: Constrained Problems

Part 3: Optimal Control Theory

Chapter 7: Optimal Control: The Maximum Principle (important chapter)

Chapter 8: More on Optimal Control

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.

Chapter 9: Infinite-Horizon Problems

Chapter 10: Optimal Control with Constraints

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.

Chapter 5: Linear Systems.

Chapter 6: Phase Plane.