📖 Course introduction

📖 Course introduction#

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Course title: Optimisation for Economics and Financial Economics

Elective second year course in the Bachelor of Economics program ECON2125
Compulsory second math course in the Master of Economics program ECON6012

The two courses are identical in content and assessment, but final grades may be adjusted depending on your program.

Course prerequisites#

See Course overview and Class summary

What you actually need to know:

basic algebra
basic calculus
some idea of what a matrix is, etc.

≈ content of EMET1001/EMET7001 math courses

Q: Is this optimization or a general math-econ course?

A: Optimization will be an important and recurring theme, but the course has a lot more material on general mathematical modeling for economics.

How tutorials will be conducted#

tutorials = practical exercises
posted with the corresponding lecture
try to solve the exercises before the tutorial
exercises solved and discussed at the tutorial
participation is not mandatory but highly recommended
if you solve all tutorial excesses, you will be able to solve all practical problems in the exams
solutions posted a week after

Tutorials start on week 2

Where to send your questions#

Administrative questions: RSE admin
- Bronwyn Cammack Senior School Administrator
- Email: enquiries.rse@anu.edu.au
- “I can not register for the tutorial group”
Content related questions: please, refer to the tutors
- “I don’t understand this step in showing that this function is continuous”
Other questions: to Fedor
- “I’m working hard but still can not keep up”
- “Can I please have extra assignment for more practice”

Attendance#

please, do not use email for instructional questions
instead make use of the office hours
office hours follow after each lecture = I will stay in the classroom to answer questions
attendance of tutorials is very highly recommended
You will make your life much easier this way
attendance of lectures is highly recommended
But not mandatory

Assessments#

Online tests, 10% each#

30 min long timed assignment
administered online through the Wattle site
one day during which to attempt this task
consists of some number of questions: multiple choice, true or false, short answer, or single numeric answer
immediate feedback
each assignment is worth 10% of your raw overall mark for this course
no late submissions will be accepted
not submitted by the due date — mark of zero
will include material from the lectures and tutorials since the last online test

Computerized quizzes are located at the course Wattle page

Individual project, 15%#

Each student will be assigned an optimization problem to solve
The problem will be of a similar type to those covered in the course and tutorials
The project will have to submitted as an explainatory video for the given problem, not longer than 5 min
The use of AI in completion of the individual project is not prohibited, but students should explain the solution using their own natural speach
The time for completion the project will be constrained to a few days
The projects will be assessed on whether the problems are solved correctly in full, clarity of explanation and quality of the presentation including technical level of the video

Grading rubric for the individual video essay#

Criterion	No points	Partial points	Full points
Correctness of the solution	Wrong solution reported	Partial answer is obtained and reported. Some inaccuracies in the derivation.	Full correct and complete solution is derived and reported. Only minor inaccuracies are permitted.
Clarity of the explanation	Explanation of the solution method and its logic is flawed. Video gives little evidence of student’s own understanding of the solution.	The video provides some evidence of understanding of the solution methods, but logic behind it is not communicated well.	The video essay provides clear evidence that the student fully understands and can communicate the logic behind the solution to the problem.
Technical level	The video and/or sound quality makes it impossible to follow the explanation of the solution.	The technical level of the video does not allow to follow the presentation of the solution clearly and in full. Video is too long.	Picture and sound quality is sufficiently high to clearly convey all the steps of the solution. Time limit is not exceeded.

Additional guidance will be provided in the second half of the course

Final exam, 45%#

Classic close book on-campus
Invigilated
3-hour exam
Only writing materials allowed
Covers the material from the whole semester!
More information will be provided by Week 10, with practice examples discussed prior to the exam
Will be held during the examination period

Note on assessments#

Exams and tests will award:

Hard work = all tutorial problems worked through
Deeper understanding of the concepts

If you follow the work done in the tutorials, you will be able to solve all the problems in the exam

In each question there will be a reasonably short path to the solution

Lectures notes/slides#

Cover exactly what you are required to know
Go back to the textbook for more details on intermediate steps
Code inserts are for illustration, they are not assessable

In particular, you need to know:

The definitions from the notes
The facts from the notes
How to apply facts and definitions

If a concept in not in the lecture notes, it is not assessable

Definitions and facts#

The lectures notes/slides are full of definitions and facts.

Definition

Functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) is called continuous at \(x\) if, for any sequence \(\{x_n\}\) converging to \(x\), we have \(f(x_n) \rightarrow f(x)\).

Possible exam question: “Show that if functions \(f\) and \(g\) are continuous at \(x\), so is \(f+g\).”

You should start the answer with the definition of continuity:

“Let \(\{x_n\}\) be any sequence converging to \(x\). We need to show that \(f(x_n) + g(x_n) \rightarrow f(x) + g(x)\). To see this, note that …”

Facts#

In the lecture notes/slides you will often see

Fact

The only \(N\)-dimensional subspace of \(\mathbb{R}^N\) is \(\mathbb{R}^N\).

This means either:

theorem
lemma
true statement taken without a proof

All well known results. You need to remember them, have some intuition for, and be able to apply.

Reading materials#

Primary reference: lecture notes
Mathematics for Economists (1994) by Simon, C. and L. Blume

[Simon and Blume, 1994]

A First Course in Optimization Theory (1996) by Rangarajan Sundaram

[Sundaram, 1996]

each lecture will reference book chapters/sections
refer to the text books for additional examples and more detailed explanations

Additional books

listed in the bibliography
additional material for self-study listed in each lecture, including additional topics and deeper coverage

Key action points for the administrative part#

Tutorials start next week, please register before the next lecture
Course content = what’s in lecture notes/slides
Lecture slides are available online and will be updated throughout the semester

What is economics? Optimization!#

Economists try to explain social phenomena in terms of the behaviour of an individual who is confronted with scarcity and the interaction of that individual with other individuals who also face scarcity. This is perhaps best captured by Malinvaud’s definition of economics:

“· · · economics is the science which studies how scarce resources are employed for the satisfaction of the needs of men living in society: on the one hand, it is interested in the essential operations of production, distribution and consumption of goods, and on the other hand, in the institutions and activities whose object it is to facilitate these operations.” (Italics in original.)

– (From page one of [Malinvaud, 1972].)

Note

A definition of economics along these lines (that is, one that emphasises the importance of scarcity) can be traced back at least as far as Lord Lionel Robbins’ justifiably famous “essay on the nature and significance of economic science”. Chapter one of this essay contains a very nice discussion of the definition of economics and its history.

The first edition of this essay was published in 1932.
The third edition of this essay was published in 1984.

The importance of constrained optimisation#

Given that scarcity is the defining feature of economics, it should not be surprising that constrained optimisation plays a central role in economic analysis. Indeed, constrained optimisation is very much a “bread and butter” skill for economists. It would difficult to make a living as an economist without some knowledge of constrained optimisation techniques.

According to [Ausubel and Deneckere, 1993] (p. 99):

“Almost every economic problem involves the study of an agent’s optimal choice as a function of certain parameters or state variables. For example, demand theory is concerned with an agent’s optimal consumption as a function of prices and income, while capital theory studies the optimal investment rule as a function of the existing capital stock.”

The three main components of mathematical economics#

Takashi Kunimoto (Unpublished lecture notes on mathematical economics, 18 May 2010, page 6) notes that, according to Rakesh Vohra (2005, Preface), the three core technical components that underlie much of economic theory are as follows.

Feasibility questions
Optimality questions
Equilibrium (fixed-point) questions

One of the main tasks of mathematical (micro-) economics is the provision of techniques to represent, analyse, and answer these three questions (and various other related questions).

Optimisation in economics: motivational quotes#

“For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear.”

– Leonhard Euler in the introduction to De Curvis Elasticis, Additamentum 1 to Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive Solutio problematis isoperimetrici latissimo sensu accepti (1744).

“Constrained-maximization problems are mother’s milk to the well-trained economist.”

– From page 88 of Caves, Richard E (1980), “Industrial organisation, corporate strategy and structure”, The Journal of Economic Literature 18(1), March, pp. 64–92.

“The very name of my subject, economics, suggests economizing or maximising. · · · So at the very foundations of our subject maximization is involved.”

– From page 249 of Samuelson, (1972), “Maximum principles in analytical economics”, The American Economic Review 62(3), June, pp. 249–262. This journal article is the text of Paul Samuelson’s Nobel Memorial Prize Lecture from 11 November 1970.