Mathematics and Economics - U6UMTHEN

In this Autumn semester module, you’ll be introduced to the basic mathematical concepts that underpin all degree programmes offered by the School of Mathematical Sciences. The major components are:

• mathematical fundamentals: logic; complex numbers; functions; set theory; introduction to cardinality; ordinary differential equations.
• linear algebra: systems of linear equations; introduction to matrices and matrix algebra.
• analysis: the real numbers; sequences; infinite series; limits and continuity of functions.
• programming in python: variables; logic and loops; functions; plotting graphs; debugging.

The overall aim of the module is to build on your existing mathematical knowledge, with an emphasis on developing mathematical skills, deepening understanding, and increasing confidence in applying a broad range of concepts and techniques. More specifically, the module introduces and provides practice in logical reasoning and rigorous mathematical thinking, particularly in relation to linear algebra and real analysis.

In this Spring semester module, you’ll build upon the basic mathematical concepts covered in ‘Core Mathematics 1’ and provides practice in logical reasoning and rigorous mathematical thinking, particularly in relation to linear algebra and real analysis. The major components are:

• linear algebra: vector spaces; linear maps; eigenvalues and eigenvectors.
• analysis: single-variable and multi-variable calculus (differential and integral).

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 modelling tools and techniques necessary to build theoretical models.

In this module you will learn about the behaviour of firms and households in situations of competitive and imperfectly competitive markets.

Probability theory 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. In this module, you will study the theory and practice of discrete and continuous probability, including topics such as Bayes’ theorem, multivariate random variables, probability distributions and the central limit theorem.

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 statistical analysis. It can be used to answer a diverse range of questions in areas such as the pharmaceuticals industry, economic planning and finance. In this module you’ll study statistical inference and learn how to analyse, interpret and report data. Topics that you’ll learn about include, point estimators and confidence intervals, hypothesis testing, linear regression and goodness-of-fit tests.

This course introduces the theory and applications of functions of a complex variable, using an approach oriented towards methods and applications. You will also learn about functions of complex variables and study topics including, analyticity, Laurent series, contour integrals and residue calculus and its applications.

In this module you will further develop your understanding of the tools of real analysis. This provides you with a solid foundation for subsequent modules in metric spaces, numerical analysis and PDE modelling. You will study topics such as convergence, norms and topology of multidimensional space, convergence of sequences of functions and applications of differentiation and integration.

Adopts a broad focus on factors influencing growth and development, concentrating on core economic policy areas and the role of international organisations, and on historical and geographical factors affecting development. Topics covered include: Economic policies, in particular the role of the IMF and the World Bank.Human development and the UN Sustainable Development Goals.Effects of aid on growth and poverty.Climate Change, poverty and conflict.Climate Change: adaptation and mitigation strategies.Trade policy and performance and the role of the WTO.Structural transformation and growthHistorical and geographical determinants of institutions and developmentRegional perspectives for East Asia, China and Africa.

This module generalises and builds upon the material covered in the year two modules, 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.  

The module adopts a chronological overview of human development and economic history from the ancient times to the 20th century. It focuses on deep rooted determinants of economic development. You’ll gain an understanding of the main themes present in the economic history debate, as well as the main methodologies employed in the field. By incorporating some of the most recent literature on the economic history of non-European countries, it provides a non-Eurocentric view of global economic growth in a historical perspective.

Topics covered include:

• pre-industrial economies
• the industrial revolution and globalisation
• the inter-war period and retreat from globalisation 
• post-WW2 economic growth and the second globalisation

This module discusses aspects of some of the main sub-areas of experimental and behavioural economics. This includes applications related to individual decision-making, strategic behaviour and market behaviour.

The module encourages reflection on both the role of experiments in economics and the assumptions that economics does (and should) make about people’s motivations. Both experimental economics and behavioural economics are still comparatively new fields within the wider discipline.

The module considers their potential and main achievements, relative to more traditional economic techniques. It encourages development of critical skills and reflection on specific research contributions in experimental and behavioural economics.

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 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
• 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.

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

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.

During this module you will build on your theoretical knowledge of statistical inference by a practical implementation of the generalised linear model. You will progress 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 module provides an introduction to coding theory in particular to error-correcting codes and their uses and applications. You’ll learn cryptography, including classical mono- and polyalphabetic ciphers.  There will also be a focus on modern public key cryptography and digital signatures, their uses and applications.

The aim of Discrete Mathematics is the study of discrete and finite rather than continuous quantities. This includes counting problems, graphs and other quantities parametrised by integers.

As such Discrete Mathematics is of great importance for various branches of Pure Mathematics, Mathematical Physics, Statistics and Computer Sciences.

The course will cover a range of Discrete Mathematics topics, including:

• an introduction to advanced aspects of combinations and permutations
• ordinary and exponential generating functions
• recurrence relations and applications to counting problems
• graphs
• digraphs
• Eulerian and Hamiltonian trails
• planar graphs
• graph colouring
• trees
• minimum spanning tree algorithms

You can choose to undertake an independent research project, involving the application of techniques of economic analysis to a research topic of your choice. For this module, you’ll write an economics research paper. You’ll be supported with writing an undergraduate dissertation in economics.

Game theory contains many branches of mathematics (and computing); the emphasis here is primarily algorithmic. The module starts with an investigation into normal-form games, including strategic dominance, Nash equilibria, and the Prisoner’s Dilemma. We look at tree-searching, including alpha-beta pruning, the ‘killer’ heuristic and its relatives. It then turns to mathematical theory of games; exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.

This module will provide an introduction to international monetary issues, including the determination of exchange rates and international spill-over effects. 

You will explore the concepts of discrete time Markov chains to understand how they used. We will also provide an introduction to probabilistic and stochastic modelling of 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 workplace applications.

This module involves the application of mathematics to a variety of practical, open-ended problems - typical of those that mathematicians encounter in industry and commerce.

Specific projects are tackled through workshops and student-led group activities. The real-life nature of the problems requires you to develop skills in model development and refinement, report writing and teamwork. There are various streams within the module, for example:

• Pure Mathematics
• Applied Mathematics
• Data Analysis
• Mathematical Physics

This ensures that you can work in the area that you find most interesting.

A 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. Key topics to be covered include:

• Vector and matrix algebra ideas relevant to multivariate statistics, including the eigen and singular value decompositions
• Methods for dimension reduction including principal components analysis, canonical correlation analysis, and factor analysis
• Multivariate regression methods.
• Classification methods, such as linear discriminant analysis.
• Multivariate normal distributions and their use in multivariate analysis of variance tests
• Exploratory data analysis methods such as clustering tools, and multi-dimensional scaling, as well as other data visualisation techniques.
• Statistical software for conducting analysis of multivariate data

This module covers the following:

Static numerical methods

• Numerical solution methods
• Numerical static optimisation methods
• Applications: resource allocation, computable general equilibrium

Dynamic numerical optimisation

• Discrete dynamic programming
• Implementation of the methods
• Applications: optimal growth, rational expectations, asset management

Agent-based economic modelling

• Foundations of agent-based modelling
• Basics of computer programming
• Applications: evolutionary games, markets   

In this module a variety of techniques of mathematical optimisation will be covered including Lagrangian methods for optimisation, simplex algorithm linear programming and dynamic programming.

These techniques have a wide range of applications to real world problems, in which a process or system needs to be made to perform optimally.

This module is concerned with the two main theories of statistical inference, namely classical (frequentist) inference and Bayesian inference.

You will explore the following topics in detail:

• sufficiency
• estimating equations
• likelihood ratio tests
• best-unbiased estimators

There is special emphasis on the exponential family of distributions, which includes many standard distributions such as the normal, Poisson, binomial and gamma.

This module 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. You will then 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. Methods for model identification for real time series data will be described. Techniques for estimating the parameters of a model, assessing its fit and forecasting future values will be developed. You will gain experience of using a statistical package and interpreting its output.