Module
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 problemsolving 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:  2021/2 
Module lecturers: 

Module credit:  15 
ECTS value:  7.5 
Prerequisites:  BEE2025 & BEE2026

Corequisites:  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 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.
ILO: Modulespecific 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: Disciplinespecific 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)
Scheduled Learning and Teaching Activities  Guided independent study  Placement / study abroad 

33  117  0 
Details of learning activities and teaching methods
Category  Hours of study time  Description 

Scheduled Learning and Teaching Activity  33  Lectures 
Guided Independent Study  117  Reading, research, reflection; preparation for lectures and assessments. 
Formative assessment
Form of assessment  Size of the assessment (eg length / duration)  ILOs assessed  Feedback method 

Homework  Weekly sets  1 8  Written and oral feedback/model solutions 
Summative assessment (% of credit)
Coursework  Written exams  Practical exams 

0  100  0 
Details of summative assessment
Form of assessment  % of credit  Size of the assessment (eg length / duration)  ILOs assessed  Feedback method 

In class tests  20  1 x 50 minutes  1  3, 78  Written and oral feedback/model solutions 
Examination  80  2 hours  4  6, 7 8  Written feedback 
Details of reassessment (where required by referral or deferral)
Original form of assessment  Form of reassessment  ILOs reassessed  Timescale for reassessment 

In class tests (20%) + Examination (80%)  Examination (2 hours, 100%)  1  8  2 hours 
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)
 Settheoretic Topology
 Fixed Points
 Sequences: Definition of Sequence and Subsequence, BolzanoWeierstrass 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 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:
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”, McGrawHill; 2^{nd} edition (2011)
 Seymour Lipschutz: “Schaums Outline of General Topology”, McGrawHill; 1 edition (2011)
 Seymour Lipschutz: “Schaum's Outline of Set Theory and Related Topics”, McGrawHill; 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”, McGrawHill; 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
18/02/2021