Advanced Mathematics for Economists

Module description

Summary:

This module is aimed at students who are considering a Masters and/or PhD in Economics. The module will cover a range of relevant mathematical tools and techniques that are typically required for further study in economics; the aim is to deepen and extend the mathematical preparation of 3rd year undergraduates by exposing them to rigorous higher level mathematics and providing them with the opportunity to develop proofs and apply new mathematical tools. Knowledge of elementary matrix theory and calculus is assumed.

The level of rigor will vary. Parts 1  and 2 will aim for thoroughness rather than for covering a huge range of material. Students should develop a feel for when a proof is complete and rigorous and when arguments are missing. In Part 3, emphasis is placed on learning how to quickly understand mathematical tools and be able to apply them, rather than on theoretical formalism. This is achieved sometimes at the expense of rigor. Where proofs are beyond the level of this course, references are given.

Additional Information:

Internationalisation
Mathematics is a global language, so the technical skills students acquire in this module can be used internationally.

Employability
By solving statistical mathematical problems and exercises, students are equipped with practical problem-solving skills, theoretical skills, and an understanding of mathematical relationships. All of these are highly valuable to employers.

Sustainability
All of the resources for this module are available on the ELE (Exeter Learning Environment).

Full module specification

Module aims

This module is aimed at students who are considering a Masters and/or PhD in Economics. The module will cover a range of relevant mathematical tools and techniques that are typically required for further study in economics; the aim is to deepen and extend the mathematical preparation of 3rd year undergraduates by exposing them to rigorous higher level mathematics and providing them with the opportunity to develop proofs and apply new mathematical tools. Knowledge of elementary matrix theory and calculus is assumed.

The level of rigor will vary. Parts 1  and 2 will aim for thoroughness rather than for covering a huge range of material. Students should develop a feel for when a proof is complete and rigorous and when arguments are missing. In Part 3, emphasis is placed on learning how to quickly understand mathematical tools and be able to apply them, rather than on theoretical formalism. This is achieved sometimes at the expense of rigor. Where proofs are beyond the level of this course, references are given.

The assessment structure on this module is subject to review and may change before the start of the new academic year. Any changes will be clearly communicated to you before the start of the term and if you wish to change module as a result of this you can do so in the module change window.

ILO: Module-specific skills

• 1. understand and be able to do basic mathematical proofs;
• 2. demonstrate the ability to work with abstract mathematical concepts;
• 3. understand the mathematical aspects of economic modelling techniques.

ILO: Discipline-specific skills

• 4. demonstrate the ability to read and work with current economic research papers;
• 5. critically analyse the logic of economic arguments;
• 6. work with economic models.

ILO: Personal and key skills

• 7. develop logic and critical thinking;
• 8. develop and deepen higher level quantitative skills.

Learning activities and teaching methods (given in hours of study time)

Details of learning activities and teaching methods

Formative assessment

Summative assessment (% of credit)

Details of summative assessment

Details of re-assessment (where required by referral or deferral)

Syllabus plan

This is an indicative outline. Further details will be available in week 1 of the module

Part 1 (3 weeks)

• Numbers
• Sets
• Proofs and mathematical logic

In Part 1 we will provide the basics for the following material. The important properties of numbers and how they are described axiomatically (in particular, their order structure and completeness) will be discussed. Central notions of set theory will be developed and illustrated. Important methods of proofs (indirect proof, inductive proof) are illustrated in examples.

Part 2 (2 weeks)

• Set-theoretic Topology
• Fixed Points
• Sequences: Definition of Sequence and Subsequence, Bolzano-Weierstrass Theorem and Cauchy criterion
• Limits and Continuity: Definition of Continuity and Uniform Continuity, Intermediate Value, Differentiation

In Part 2 we will give a rigorous introduction to basic topological concepts (limit points, neighbourhoods, compact spaces, metric spaces etc.) and provide many examples of topological spaces. The emphasis is on teaching how to do rigorous proofs using abstract concepts.

Part 3 (5 weeks)

• Ordinary differential equations
• Eigenvectors and Eigenvalues
•  First order differential equations systems:
• Other applications of mathematics to economics, such as discrete math and combinatorics.

In Part 3 we will introduce the most elementary notions of first and second differential equations. We will subsequently study systems of linear differential equations, their solutions in the various cases, stability conditions, phase diagrams and some economic applications. For this part only elementary matrix theory and basic calculus are needed.

Indicative learning resources - Basic reading

Basic reading:

Parts 1 and 2: Logic, set theory, topology

• Dr Kevin Houston: “How to Think Like a Mathematician: A Companion to Undergraduate Mathematics”, Cambridge University Press (2009)
• John Nolt, Dennis Rohatyn and Achille Varzi: “Schaum's Outline of Logic”, McGraw-Hill; 2^nd  edition (2011)
• Seymour Lipschutz: “Schaums Outline of General Topology”, McGraw-Hill; 1 edition (2011)
• Seymour Lipschutz: “Schaum's Outline of Set Theory and Related Topics”, McGraw-Hill; 2 edition (July 1, 1998) 

(The emphasis will be on the first two books. Exact edition is not important.)

Part 3: Differential Equations and Dynamic Optimisation

• Edward Dowling: “Schaum's Outline of Introduction to Mathematical Economics”, McGraw-Hill; 3 edition (2011)  
• Knut Sydsaeter, Peter Hammond, Atle Seierstad, Arne Strom: “Further Mathematics for Economic Analysis” Prentice Hall; 2 edition (2008)

Module has an active ELE page?

Yes

Origin date

01/09/2011

Last revision date

01/03/2022