Undergraduate Course: Advanced Mathematical Economics (ECNM10085)

Course Outline
School	School of Economics	College	College of Arts, Humanities and Social Sciences
Credit level (Normal year taken)	SCQF Level 10 (Year 3 Undergraduate)	Availability	Available to all students
SCQF Credits	20	ECTS Credits	10
Summary	This course is about the advanced mathematical tools that are used in economics research. Each mathematical topic is explored in the context of an important economic problem.
Course description	The topics covered vary from year to year. An example curriculum would be the following mathematics concepts illustrated in the context of general equilibrium theory: * Naive Set Theory. This is the language of mathematics, and is widely used by economists. This is important for making precise hypotheses, such as "in every equilibrium, real wages increase over time", and for verifying these hypotheses with logically sound proofs. The main concepts are: sets, functions, logical connectives, quantifiers, countability, induction, proof by contradiction. * Real Analysis and Metric Spaces. This branch of mathematics focuses on continuity and nearness (topology) while putting geometric concepts like distance and angles into the background. These ideas are useful for determining whether an optimal decision is possible, whether an equilibrium of an economy exists, and determining when optimal decisions change drastically when circumstances change. The main concepts are: open sets, continuity, limits, interior, boundary, closure, function spaces, sup metric, Cauchy sequences, connected spaces, complete spaces, compact spaces, Bolzano-Weierstrass theorem, Banach fixed point theorem, Brouwer fixed point theorem. * Convex Analysis. This branch of geometry focuses on comparing extreme points and intermediate points that lie between extremes. These tools are useful for determining whether there is one or several optimal decisions in a particular situation, and determining in which direction optimal choices move when circumstances change. Convex analysis is related to the economic notions of increasing marginal cost and decreasing marginal benefit. The main concepts are: convex sets, convex and concave functions, quasi-convex and quasi-concave functions, supporting hyperplane theorem, separating hyperplane theorem. * Dynamic Programming. This branch of mathematics is about breaking up a complicated optimisation problem involving many decisions into many simple optimisation problems involving few decisions. For example, a lifetime of choices can be broken up into simple choices made day-by-day. The main concepts are: value functions, Bellman equations, Bellman operators. * Envelope Theorem. This is a calculus formula for calculating marginal values, such marginal benefit of saving money. The main concepts are: differentiable support functions, the Benveniste-Scheinkman theorem.

Entry Requirements (not applicable to Visiting Students)
Pre-requisites	Students MUST have passed: ( Economics 2 (ECNM08006) AND Statistical Methods for Economics (ECNM08016)) OR ( Probability (MATH08066) AND Statistics (Year 2) (MATH08051)) OR Data Analysis for Psychology in R 2 (PSYL08015)	Co-requisites
Prohibited Combinations		Other requirements	Continuing Professional Development students should usually have an equivalent of at least 4 semester-long Economics courses at grade B or above (or be predicted to obtain this) for entry to this course. This MUST INCLUDE courses in Intermediate Macroeconomics (with calculus); Intermediate Microeconomics (with calculus); and Probability and Statistics. If macroeconomics and microeconomics courses are not calculus-based, then, in addition, Calculus (or Mathematics for Economics) is required. We will only consider University/College level courses.

Information for Visiting Students
Pre-requisites	Visiting students must have an equivalent of at least 4 semester-long Economics courses at grade B or above for entry to this course. This MUST INCLUDE courses in Intermediate Macroeconomics (with calculus); Intermediate Microeconomics (with calculus); and Probability and Statistics. If macroeconomics and microeconomics courses are not calculus-based, then, in addition, Calculus (or Mathematics for Economics) is required.
High Demand Course?	Yes

Course Delivery Information

Academic year 2022/23, Available to all students (SV1)		Quota: None
Course Start	Semester 1
Timetable	Timetable
Learning and Teaching activities (Further Info)	Total Hours: 200 ( Lecture Hours 20, Seminar/Tutorial Hours 18, Summative Assessment Hours 6, Programme Level Learning and Teaching Hours 4, Directed Learning and Independent Learning Hours 152 )
Assessment (Further Info)	Written Exam 90 %, Coursework 10 %, Practical Exam 0 %
Additional Information (Assessment)	Coursework (10%) Degree exam (90%). Coursework involves weekly homeworks. There would be homework due each week (except the first), and students would get the full 10% if they attempt at least 6 of the 9, and lose 2% for each subsequent homework missed. There will be two exams: one in the December Diet and one in the April/May Diet. Candidates will be awarded the maximum of the marks obtained in these exams. Part-year visiting students only (VV1) AND Continuing Professional Development students - Weekly homework 10% - Mathematical Economics Project 40% (optional) - 3 Hour Examination in December 50% (90% of the optional project is not undertaken) Coursework involves weekly homeworks. There would be homework due each week (except the first), and students would get the full 10% if they attempt at least 6 of the 9, and lose 2% for each subsequent homework missed. The Mathematical Economics Project is worth 40%, but it is optional and only counts towards the final grade if this is favourable to the student. The examination in December is three hours. - - - - - While we recommend that most Continuing Professional Development students take this as a full-year course, this course is also available in a one-semester format. For our internal record-keeping purposes, we call this option 'part-year visiting student' (because we offer the same format to exchange students), even though this is a Continuing Professional Development course.
Feedback	It is very important that students attend all tutorials, to receive feedback about their work and thinking.
Exam Information
Exam Diet	Paper Name		Hours & Minutes
Main Exam Diet S1 (December)			3:00
Main Exam Diet S2 (April/May)			3:00

Academic year 2022/23, Part-year visiting students only (VV1)		Quota: None
Course Start	Semester 1
Timetable	Timetable
Learning and Teaching activities (Further Info)	Total Hours: 200 ( Lecture Hours 20, Seminar/Tutorial Hours 18, Summative Assessment Hours 3, Programme Level Learning and Teaching Hours 4, Directed Learning and Independent Learning Hours 155 )
Assessment (Further Info)	Written Exam 50 %, Coursework 50 %, Practical Exam 0 %
Additional Information (Assessment)	Coursework (10%) Degree exam (90%). Coursework involves weekly homeworks. There would be homework due each week (except the first), and students would get the full 10% if they attempt at least 6 of the 9, and lose 2% for each subsequent homework missed. There will be two exams: one in the December Diet and one in the April/May Diet. Candidates will be awarded the maximum of the marks obtained in these exams. Part-year visiting students only (VV1) AND Continuing Professional Development students - Weekly homework 10% - Mathematical Economics Project 40% (optional) - 3 Hour Examination in December 50% (90% of the optional project is not undertaken) Coursework involves weekly homeworks. There would be homework due each week (except the first), and students would get the full 10% if they attempt at least 6 of the 9, and lose 2% for each subsequent homework missed. The Mathematical Economics Project is worth 40%, but it is optional and only counts towards the final grade if this is favourable to the student. The examination in December is three hours. - - - - - While we recommend that most Continuing Professional Development students take this as a full-year course, this course is also available in a one-semester format. For our internal record-keeping purposes, we call this option 'part-year visiting student' (because we offer the same format to exchange students), even though this is a Continuing Professional Development course.
Feedback	It is very important that students attend all tutorials, to receive feedback about their work and thinking.
Exam Information
Exam Diet	Paper Name		Hours & Minutes
Main Exam Diet S1 (December)			3:00
Resit Exam Diet (August)			3:00

Learning Outcomes
On completion of this course, the student will be able to: Mathematical maturity, i.e. the ability to: distinguish between definitions, conjectures, theorems, and proofs, generalise and specialise theorems and proofs, devise counter-examples, and determine whether objects conform to definitions and conditions of theorems. Experience in applying mathematical tools to derive economic conclusions. Research and investigative skills such as problem framing and solving and the ability to assemble and evaluate complex evidence and arguments. Communication skills in order to critique, create and communicate understanding and to collaborate with and relate to others. Personal effectiveness through task-management, time-management, teamwork and group interaction, dealing with uncertainty and adapting to new situations, personal and intellectual autonomy through independent learning. Practical/technical skills such as, modelling skills (abstraction, logic, succinctness), qualitative and quantitative analysis and general IT literacy.

Reading List
Indicative readings: * Boyd and Vandenburghe (2004), "Convex Optimization", Cambridge University Press. * Luenberger (1968), "Optimization by Vector Space Methods", Wiley. * de la Fuente (2000), "Mathematical Methods and Models for Economists", Cambridge University Press.

Additional Information
Graduate Attributes and Skills	Research and Inquiry B1. The ability to identify, define and analyse theoretical and applied economic problems and identify or devise approaches to investigate and solve these problems. B3. The ability to critically assess existing understanding of economic and social issues, the limitations of that understanding and the limitations of their own knowledge and understanding of those issues. B4. The ability to question the principles, methods, standards and boundaries of economic knowledge Personal and Intellectual Autonomy C1. The ability to be independent learners who take responsibility for their own learning, and are committed to continuous reflection, self-evaluation and self-improvement. C4. The ability to collaborate and debate effectively to test, modify and strengthen their own views. Communication D1. The ability to make effective use of oral, written and visual means to critique, create and communicate understanding. D2. The ability to further their own learning through effective use of feedback. D3. The ability to use communication as a tool for collaborating and relating to others. Personal Effectiveness E1. The ability to manage tasks and also skills in time-management. E4. The ability to work effectively with others, capitalising on their different thinking.
Additional Class Delivery Information	10 * 2 hour lectures 9 * 2 hour tutorials
Keywords	AdvMath