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.
The level of rigor will vary. Parts 2 - 4 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 1 emphasis is placed on a clear and non-technical presentation of the various technical concepts and their applications, 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 and plain explanation of the various points of the theorems is always provided and the ways they are used are always indicated.

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 title:Advanced Mathematics for Economists
Module code:BEE3054
Module level:3
Academic year:2016/7
Module lecturers:
  • Professor Dieter Balkenborg - Convenor
Module credit:15
ECTS value:

7.5

Pre-requisites:

BEE1024; BEE2025 & BEE2026 or BEE2024

Co-requisites:

None

Duration of module: Duration (weeks) - term 2:

11

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.
The level of rigor will vary. Parts 2 - 4 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 1 emphasis is placed on a clear and non-technical presentation of the various
technical concepts and their applications, 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 and plain explanation of the various points of the theorems is always provided and the ways they are used are always indicated.

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 analyze 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)

Scheduled Learning and Teaching ActivitiesGuided independent studyPlacement / study abroad
331170

Details of learning activities and teaching methods

CategoryHours of study timeDescription
Contact hours33Lectures

Formative assessment

Form of assessmentSize of the assessment (eg length / duration)ILOs assessedFeedback method
Homework Bi-weekly sets1 -8Written and oral feedback / model solutions

Summative assessment (% of credit)

CourseworkWritten examsPractical exams
01000

Details of summative assessment

Form of assessment% of creditSize of the assessment (eg length / duration)ILOs assessedFeedback method
In class tests202 x 40 mins1 - 3, 7-8Written and oral feedback / model solutions
Examination802 hours4 - 6, 7 -8Written feedback
0
0
0
0

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

Original form of assessmentForm of re-assessmentILOs re-assessedTimescale for re-assessment
Examination / in class tests (100%)Examination1 - 82 hours

Syllabus plan

Part 1 (2 weeks)
• Numbers
• Sets
• Proofs
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 (3 weeks)
• Set-theoretic Topology
• Fixed Points
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. We provide an elementary geometric proof of Brouwer’s fixed point theorem and illustrate the Poincare index theorem for the plane. The emphasis is on teaching how to do rigorous proofs using abstract concepts.

Part 3 (3 weeks)
• Sequences: Definition of Sequence and Subsequence, Bolzano-Weierstrass Theorem and Cauchy criterion
• Limits and Continuity: Definition of Continuity and Uniform Continuity, Intermediate Value
• Differentiation: Definition, Theorems of Rolle, Lagrange, L’Hôpital and Cauchy
• Elements of Integration in Rn: Definition, Fundamental Theorem of Calculus, Differentiation of the Integral (Leibniz Rule).
In Part 3, we will give a rigorous introduction to the main concepts and theorems of basic mathematical analysis. The focus is on the foundational theorems like the intermediate value theorem, Rolle’s theorem, Fundamental Theorem of Calculus, Bolzano-Weierstrass. The student is expected to solve simple limits, derivatives and integrals using first principles and definitions, etc.

Part 4 (3 weeks)
• Ordinary differential equations
• Eigenvectors and Eigenvalues
• First order differential equations systems: Stability and bifurcation analysis
In Part 4 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:

Part 1: 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; 2nd  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 2: Differential Equations and Dynamic Optimization

Module has an active ELE page?

Yes

Origin date

01/09/2011

Last revision date

01/08/2013