Mathematics and Economics BSc

The idea of proof is fundamental to all mathematics. We’ll look at mathematical reasoning using techniques from logic to deal with sets, functions, sequences and series.

This module links directly with your study in Calculus and Linear Mathematics. It provides you with the foundations for the broader area of Mathematical Analysis. This includes the rigorous study of the infinite and the infinitesimal.

You will also learn the basics of computer programming. This will give you the chance to use computational algorithms to explore many of the mathematical results you’ll encounter in your core modules.

Your study will include:

• propositional and predicate logic; set theory, countability
• proof: direct, indirect and induction
• sequences and infinite series (convergence and divergence)
• limits and continuity of functions
• programming in Python

How do we define calculus? How is it used in the modern world?

The concept can be explained as the mathematics of continuous change. It allows us to analyse motion and change in time and space.

You will cover techniques for differentiating, integrating and solving differential equations. You’ll learn about the theorems which prove why calculus works. We will explore the theory and how it can be applied in the real world.

Your study will include:

• functions: limits, continuity and differentiability, rules of differentiation
• techniques for integration, fundamental theorem of calculus
• solution of linear and nonlinear differential equations
• multivariate calculus, Lagrange multipliers, stationary points
• multiple integrals, changes of variables, Jacobians

This module gives you the mathematical tools required for later modules which involve modelling with differential equations. These include:

• mathematical physics
• mathematical medicine and biology
• scientific computation

Vectors, matrices and complex numbers are familiar topics from A level Mathematics and Further Mathematics. Their common feature is linearity. A linear mathematical operation is one which is compatible with addition and scaling.

As well as these topics you’ll study the concept of a vector space, which is fundamental to later study in abstract algebra. We will also investigate practical aspects, such as methods for solving linear systems of equations.

The module will give you the tools to analyse large systems of equations that arise in mathematical, statistical and computational models. For example, in areas such as:

• fluid and solid mechanics
• mathematical medicine and biology
• mathematical finance

Your study will include:

• complex numbers, vector algebra and geometry
• matrix algebra, inverses, determinants
• vector spaces, subspaces, bases
• linear systems of simultaneous equations, Gaussian elimination
• eigenvalues and eigenvectors, matrix diagonalisation
• linear transformations, inner product spaces

What is the importance of probability in the modern world?

It allows us to assess risk when calculating insurance premiums. It can help when making investment decisions. It can be used to estimate the impact that government policy will have on climate change or the spread of disease.

We will look at the theory and practice of discrete and continuous probability. Your study will include:

• sample spaces, events and counting problems
• conditional probability, independence, Bayes’ theorem
• random variables, expectation, variance
• discrete and continuous probability distributions
• multivariate random variables
• sums of random variables, central limit theorem

These topics will help you prepare for later modules in:

• probability methods
• stochastic models
• uncertainty quantification
• mathematical finance

Statistics is concerned with methods for collecting, organising, summarising, presenting and analysing data. It enables us to draw valid conclusions and make reasonable decisions based on the analysis. It can be used to answer a diverse range of questions such as:

• Do the results of a clinical trial indicate that a new drug works?
• Is the HS2 rail project likely to be cost-effective?
• Should a company lend money to a customer with a given credit history?

In this module you’ll study statistical inference and learn how to analyse, interpret and report data. You’ll learn about the widely used statistical computer language R.

Your study will include:

• exploratory data analysis
• point estimators, confidence intervals
• hypothesis testing
• correlation, statistical inference
• linear regression, chi-squared tests

These first-year topics give you the foundations for later related modules in:

• statistical models and methods
• data analysis and modelling
• statistical machine learning

The first semester provides an introduction to microeconomics, including behaviour of firms and households in situations of competitive and imperfectly competitive markets. The second semester provides an introduction to macroeconomics.

Macroeconomics is the study of the aggregate economy, focusing on the cyclical pattern of aggregate output and co-movement of real and monetary aggregates in general equilibrium. A series of basic models used in modern macroeconomics are introduced, with a particular focus on dynamic general equilibrium modeling tools and techniques necessary to build theoretical models.

How can the flight-path of a spacecraft to another planet be planned? How many fish can we catch without depleting the oceans? How long would it take a lake to recover after its pollution is stopped?

The real world is often too complicated to get exact information. Instead, mathematical models can help by providing estimates. In this module, you’ll learn how to construct and analyse differential equations which model real-life applications.

Your study will include:

• modelling with differential equations
• kinematics and dynamics of moving bodies
• Newton’s laws, balance of forces
• oscillating systems, springs, simple harmonic motion
• work, energy and motion

You'll be able to expand on these techniques later in your degree through topics such as:

• black holes, quantum theory
• fluid and solid mechanics
• mathematical medicine and biology
• mathematical finance

In this module you will learn about the theory and applications of functions of a complex variable using a method and applications approach. You will develop an understanding of the theory of complex functions and evaluate certain real integrals using your new skills.

This module is a general introduction to the economic problems of developing countries. The module will cover such topics as:

• the implications of history and expectation
• poverty, income distribution and growth
• fertility and population
• employment, migration and urbanisation
• markets in agriculture
• agricultural household models
• risk and insurance
• famines

This course is an introduction to Fourier series and integral transforms and to methods of solving some standard ordinary and partial differential equations which occur in applied mathematics and mathematical physics.

The course describes the solution of ordinary differential equations using series and introduces Fourier series and Fourier and Laplace transforms, with applications to differential equations and signal analysis. Standard examples of partial differential equations are introduced and solution using separation of variables is discussed.

This module will offer an introduction to some theoretical concepts related to the allocation of risk by financial institutions. Then it will apply these concepts to the analysis of financial and banking crises.

Pure mathematics at university is typically very different to the pure mathematics you've learnt at school or college. You'll use the language of sets, functions and relations to study some very abstract mathematical ideas.

In this module, we'll develop the skills of reading and writing the language of pure mathematics. You will learn techniques to build mathematical proofs in an abstract setting.

Your study will include:

• the language of set theory
• relations and functions
• rational and irrational numbers
• modular arithmetic
• prime factorisation

These topics will provide you with the basics you need for subsequent modules in algebra, number theory and group theory.

This module provides an advanced economic analysis of the theory of organisation of firms and industries. It will analyse a variety of market structures related to the degree of market competition with a special emphasis on imperfectly competitive markets. It will also analyse issues related to the internal organisation of firms.

This module is an introduction to international trade theory and policy. It covers the core trade theories under perfect and imperfect competition and applies them to understanding the pattern of trade, gains from trade and modern topics like foreign outsourcing. On the policy side, it examines the effects of different government trade policy instruments and the role of international trade agreements.

This module is concerned with the effect of political and institutional factors on economic variables as well as with the study of politics using the techniques of economics.

This module introduces basic techniques in numerical methods and numerical analysis which can be used to generate approximate solutions to problems that may not be amenable to analysis. Specific topics include:

• Implementing algorithms in Matlab
• Discussion of errors (including rounding errors)
• Iterative methods for nonlinear equations (simple iteration, bisection, Newton, convergence)
• Gaussian elimination, matrix factorisation, and pivoting
• Iterative methods for linear systems, matrix norms, convergence, Jacobi, Gauss-Siedel
• Interpolation (Lagrange polynomials, orthogonal polynomials, splines)
• Numerical differentiation & integration (Difference formulae, Richardson extrapolation, simple and composite quadrature rules)
• Introduction to numerical ODEs (Euler and Runge-Kutta methods, consistency, stability) 

This module provides an introduction to the economics of the labour market. We will look at some basic theories of how labour markets work and examine evidence to see how well these theories explain the facts.

Particular attention will be given to the relationship between the theory, empirical evidence and government policy. The module will refer especially to the UK labour market, but reference will also be made to other developed economies.

This module will address both the fundamental and applied aspects of macroeconomic theory. In particular, the module will focus on:

• introducing the modern theory of expectations and economic dynamics
• using this approach to think about short run fluctuations
• studying the role of macro policy on short run fluctuations

The module will review the so-called modern approach to aggregate demand and aggregate supply. This entails incorporating into the classical approach to aggregate supply and aggregate demand, insights from Keynesian economics. This will serve as a base to discuss the role of macro policy in controlling for fluctuations in output and employment. 

In this module you will build on the foundation of knowledge gained from your core year one modules in Analytical and Computational Foundations and Calculus. You will learn to follow a rigorous approach needed to produce concrete proof of your workings.

Groups, rings and fields are abstract structures which underpin many areas of mathematics. For example, addition of integers fits the structure of a group. However, by analysing the general concept of a group, our proofs are relevant to many other areas of mathematics.

You will build on your understanding of Foundations of Pure Mathematics. Together we will develop a deeper knowledge of abstract algebraic structures, particularly groups. This provides the foundation for subsequent modules in abstract algebra and number theory.

Your study will include:

• symmetries
• groups, cyclic groups, Lagrange’s theorem
• rings and fields
• integer arithmetic, Euclid’s algorithm
• polynomial arithmetic, factorisation

This module covers intermediate microeconomics including general equilibrium analysis; welfare economics; elementary game theory; and strategic behaviour of firms.

This module will provide a foundation for the monetary economics modules in the third year and is a complement to financial economics for the second and third years. It will cover topics such as the definitions and role of money, portfolio choice, financial markets and banks, central banks and monetary policy, and the monetary transmission mechanism. 

Under these headings the module will address issues of theory, policy and practice relating to recent experience in the UK and other countries. The module will feature some current debates and controversies based on recent events.

This module will give you an introduction to the theory of probability and random variables, with particular attention paid to continuous random variables. Fundamental concepts relating to probability will be discussed in detail, including well-known limit theorems and the multivariate normal distribution. You will then progress onto complex topics such as transition matrices, one-dimensional random walks and absorption probabilities.

This module looks at:

• public finances in the UK
• market failures
• fundamental theorems of welfare economics
• social welfare functions
• externalities
• public goods
• natural monopolies
• public choice
• social insurance: social security, taxation and equity
• excess burden of taxation and tax incidence

The first part of this module provides an introduction to statistical concepts and methods and the second part introduces a wide range of techniques used in a variety of quantitative subjects. The key concepts of inference including estimation and hypothesis testing will be described as well as practical data analysis and assessment of model adequacy.

This course aims to give students a sound grounding in the application of both differential and integral calculus to vectors, and to apply vector calculus methods and separation of variables to the solution of partial differential equations. The module is an important pre-requisite for a wide range of other courses in Applied Mathematics.

The module introduces you to a range of statistical techniques that can be used to analyse the characteristics of univariate economic time series. The basic theoretical properties of time series models are discussed and we consider methods for fitting and checking the adequacy of empirical time series models. Methods of forecasting future values of economic time series are then considered. If reassessment is required, a single examination will replace all failed assessment components of the module.

An independent research project, involving the application of techniques of economic analysis to a self-chosen research topic and the presentation of a written report. There will be lectures to provide general guidance on economic research methods and writing an undergraduate dissertation in economics.

Topics include:

• introduction to the dissertation
• types of dissertation
• literature reviews
• sources of data
• writing up your dissertation
• data entry and data management
• an introduction to STATA
• descriptive statistics
• practical issues in regression analysis
• model selection
• endogeneity bias

This module generalises and builds upon the material covered in the Econometric Theory I and II. In the first part of the module, we study large sample, or asymptotic, theory. This is needed in order to obtain tractable results about the behaviour of estimators and tests when the standard modelling assumptions - which frequently cannot be verified in practice - are relaxed.

The second part of the module continues the time series analysis taken in Econometric Theory II, with the emphasis on the behaviour of typical economic time series, and the implications of that behaviour in practical analysis, such as the construction of models linking economic time series. The key issues addressed will be the identification of non-stationarity through the construction of formal tests and the implications for modelling with non-stationary data.

Particular attention will be paid to the contributions of Sir Clive Granger to the spurious regression problem and to cointegration analysis, for which he was ultimately awarded the Nobel Prize.

This module covers:

• saving, focusing on how agents make intertemporal decisions about their savings and wealth accumulation
• saving puzzles and household portfolios, focusing on credit markets and credit markets' imperfections, and why do households hold different kinds of assets
• asset allocation and asset pricing, focusing on intertemporal portfolio selection, asset pricing and the equity premium puzzle
• bond markets and fixed income securities
• the term structure of interest rates
• the role of behavioural finance in explaining stock market puzzles

This module looks at:

• trade policy
• economic policy for trade and international factor mobility
• theory and evidence
• trade policy and imperfect competition
• trade and distortions
• the political economy of protection
• trade policy reform

This module covers an economic analysis of the labour market, with an emphasis on policy implications and institutional arrangements.

This module covers:

• dynamic general equilibrium models, focusing on how the time path of consumption, and saving, is determined by optimising agents and firms that interact on competitive markets
• growth in dynamic general equilibrium, focusing on the Solow model and the data, and the role played by accumulation of knowledge (endogenous innovation) in explaining long run growth
• Real Business Cycles (RBC), focusing on how the RBC approach accounts for business cycle fluctuations, and what links short run fluctuations and growth processes

This module is intended to provide an introduction to mathematical techniques used in economics. In particular, examples of economic issues that can be analysed using mathematical models will be discussed in detail.

Particular attention will be given to providing an intuitive understanding of the logic behind the formal results presented.

This module will cover topics in advanced microeconomics and decision theory. The precise content may vary from year to year, but the module will start from the basis established by the Microeconomic Theory module.

This module provides a rigorous introduction to formal models of money in the macroeconomy. Following this, applications for areas of central banking, finance and international macroeconomics will be explored.

The module will introduce some major themes of the economic analysis of government. Using the tools of modern microeconomic theory, it will explore how government institutions are designed, how they could be designed better, and how they shape economic policy.

This module is a continuation of the module on time series analysis. While the earlier module was devoted to basic time series model building methodology, this module concentrates on those developments which can be applied in the subject of economics. 

In particular, the emphasis will be on aspects of the behaviour of typical economic time series, and the implications of that behaviour in practical analysis, such as the construction of models linking economic time series.

The key issues addressed will be the identification of non-stationarity through the construction of formal tests and the implications for modelling with non-stationary data. Particular attention will be paid to the contributions of Sir Clive Granger to the spurious regression problem and to cointegration analysis, for which he was ultimately awarded the Nobel Prize.

This module provides an advanced economic analysis of the theory of organisation of firms and industries. It will analyse a variety of market structures related to the degree of market competition with a special emphasis on imperfectly competitive markets. It will also analyse issues related to the internal organisation of firms.

This module looks at:

• trade policy - theory and evidence
• trade policy and imperfect competition
• trade and distortions
• the political economy of protection
• trade policy reform

This module focuses on a range of econometric methods used in policy evaluation and in the identification and estimation of causal effects. Topics to be covered include:

• potential outcomes framework
• regression analysis and matching
• instrumental variables
• difference-in-differences
• regression discontinuity

This module covers: 

• Foundations:
  • The rational political individual?
  • Voter participation
  • Collective action and the role of the state
• Core Political Economy:
  • The economic approach to politics
  • Political aspects of economics: rights and the limits of the state
  • Political aspects of economics: inequality and the duties of the state
• Political Economy in Action:
  • Political economy in action: some current issues in political economy

In this module you will build on your theoretical knowledge of statistical inference by a practical implementation of the generalised linear model. You will move on to enhance your understanding of statistical methodology including the analysis of discrete and survival data. You will also be trained in the use of a high-level statistical computer program.

This course provides an introduction to coding theory in particular to error-correcting codes and their uses and applications. It also provides an introduction to to cryptography, including classical mono- and polyalphabetic ciphers as well as modern public key cryptography and digital signatures, their uses and applications.

A graph (in the sense used in Graph Theory) consists of vertices and edges, each edge joining two vertices. Graph Theory has become increasingly important recently through its connections with computer science and its ability to model many practical situations. 

Topics covered in the course include:

• paths and cycles
• the resolution of Euler’s Königsberg Bridge Problem
• Hamiltonian cycles
• trees and forests
• labelled trees,
• the Prüfer correspondence
• planar graphs
• Demoucron et al. algorithm
• Kruskal's algorithm
• the Travelling Salesman's problem
• the statement of the four-colour map theorem
• colourings of vertices
• chromatic polynomial
• colourings of edges.

This module gives an introduction into some basic ideas of functional analysis with an emphasis on Hilbert spaces and operators on them.

Many concepts from linear algebra in finite dimensional vector spaces (e.g. writing a vector in terms of a basis, eigenvalues of a linear map, diagonalisation etc.) have generalisations in the setting of infinite dimensional spaces making this theory a powerful tool with many applications in pure and applied mathematics

In this module the concepts of discrete time Markov chains are explored and used to provide an introduction to probabilistic and stochastic modelling for investment strategies, and for the pricing of financial derivatives in risky markets. You will gain well-rounded knowledge of contemporary issues which are of importance in research and applications.

Metric space generalises the concept of distance familiar from Euclidean space. It provides a notion of continuity for functions between quite general spaces.

The module covers metric spaces, topological spaces, compactness, separation properties like Hausdorffness and normality, Urysohn’s lemma, quotient and product topologies, and connectedness. Finally, Borel sets and measurable spaces are introduced.

This module is concerned with the analysis of multivariate data, in which the response is a vector of random variables rather than a single random variable. A theme running through the module is that of dimension reduction. Key topics to be covered include: principal components analysis, whose purpose is to identify the main modes of variation in a multivariate dataset; modelling and inference for multivariate data, including multivariate regression data, based on the multivariate normal distribution; classification of observation vectors into sub-populations using a training sample; canonical correlation analysis, whose purpose is to identify dependencies between two or more sets of random variables. Further topics to be covered include factor analysis, methods of clustering and multidimensional scaling.

In this module a variety of techniques and areas of mathematical optimisation will be covered including Lagrangian methods for optimisation, simplex algorithm linear programming and dynamic programming. You’ll develop techniques for application which can be used outside the mathematical arena. 

In this module you will explore two main concepts of statistical inference; classical (frequentist) and Bayesian. Topics such as sufficiency, estimating equations, likelihood ratio tests and best-unbiased estimators will be discussed in detail. You will gain knowledge of the theory and concepts underpinning contemporary research in statistical inference and methodology.

In this module you will develop your knowledge of discrete-time Markov chains by applying them to a range of stochastic models. You will be introduced to Poisson and birth-and-death processes and then you will move onto more extensive studies of epidemic models and queuing models with introductions to component and system reliability.

This module will provide a general introduction to the analysis of data that arise sequentially in time. Several commonly-occurring models will be discussed and their properties derived, along with methods for model identification for real-time series data. You will develop techniques for estimating the parameters of a model, assessing its fit and forecasting future values. You will also gain experience of using a statistical package and interpreting its output.

This module consists of a self-directed investigation of a project selected from a list of projects or, subject to prior approval of the School, from elsewhere.

Project modules are carried out in the Autumn and Spring semesters.

The project will be supervised by a member of staff and will be based on a substantial mathematical problem, an application of mathematics or investigation of an area of mathematics not previously studied by the student. The course includes training in the use of IT resources, the word-processing of mathematics and report writing.