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 nontechnical 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 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:  2016/7 
Module lecturers: 

Module credit:  15 
ECTS value:  7.5 
Prerequisites:  BEE1024; BEE2025 & BEE2026 or BEE2024 
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.
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 nontechnical 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: 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 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 Activities  Guided independent study  Placement / study abroad 

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

Contact hours  33  Lectures 
Formative assessment
Form of assessment  Size of the assessment (eg length / duration)  ILOs assessed  Feedback method 

Homework  Biweekly 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  2 x 40 mins  1  3, 78  Written and oral feedback / model solutions 
Examination  80  2 hours  4  6, 7 8  Written feedback 
0  
0  
0  
0 
Details of reassessment (where required by referral or deferral)
Original form of assessment  Form of reassessment  ILOs reassessed  Timescale for reassessment 

Examination / in class tests (100%)  Examination  1  8  2 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)
• Settheoretic 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, BolzanoWeierstrass 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, BolzanoWeierstrass. 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”, 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 2: Differential Equations and Dynamic Optimization
 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
01/08/2013