BSc (Hons) Mathematics with Economics (Placement Year) Bachelor's degree at Lancaster University

Interested in how mathematicians build theories from basic concepts to complex ideas, like eigenvalues and integration? Journey from polynomial operations to matrices and calculus through this module. Starting with polynomials and mathematical induction, you will learn fundamental proof techniques. You will explore matrices, arrays of numbers encoding simultaneous linear equations, and their geometric transformations, which are essential in linear algebra. Eigenvalues and eigenvectors, which characterise these transformations, will be introduced, highlighting their role in applications including population growth and Google's page rankings. Next, we will reintroduce you to calculus, from its

This module provides a comprehensive introduction to macroeconomics, which involves the study of economics at an aggregate level. We will cover various topics, including national income analysis, monetary theory, business cycles, inflation, unemployment, and the great macroeconomic debates. The module provides the foundations for further study in Economics. Throughout the module, we will develop essential theoretical concepts and demonstrate how they apply to real-world situations. The module is self-contained and can be taken by students with no prior knowledge of macroeconomics. It takes a more mathematical approach to the subject than Foundations of Macroeconomics.

You will receive a thorough introduction to microeconomics, which is the analysis of Economics at the level of the individual or firm. The topics you will cover include the theory of demand and supply, costs and pricing under various forms of market structure, and welfare economics. The module lays the groundwork for further study in Economics. In addition to developing key theoretical concepts, we will illustrate how these concepts can be applied to real-world examples. The module is self-contained and is suitable for students without prior knowledge of the subject. This module provides a more mathematical treatment of microeconomics than Foundations of Microeconomics.

An introduction to the mathematical and computational toolsets for modelling the randomness of the world. You will learn about probability, the language used to describe random fluctuations, statistics and the mathematical techniques used to extract meaning from data. You will explore how computing tools can be used to solve challenges in scientific research, artificial intelligence, machine learning and data science. You will develop the axiomatic theory of probability, discover the theory and uses of random variables and investigate how theory matches intuitions about the real-world. You will then dive into statistical inference, learning to select appropriate probability models to describ

At university, emphasis is placed on understanding general mathematical theorems. They apply in many different cases, and understanding why a result is true enables us to creatively use the underlying ideas to tackle new problems. Study the language and structure of mathematical proofs, illustrated by results from number theory. You will see the concept of congruence of integers, which is a simplified form of arithmetic where seemingly impossible problems become solvable. In relation, you’ll encounter the abstract idea of an equivalence relation. Sets and functions form the basic language of mathematics. You will study functions of a real variable and abstract functions between arbitrary set

A mathematical model is a representation of a real-world event, such as a building vibrating during an earthquake or the spread of a disease within a population. In this module, you will investigate mathematical models that lead to ordinary differential equations and will study a variety of core analytical methods for solving them, such as integrating factors and separation of variables. You will learn to develop models by extracting important information from real-world scenarios, which can then be analysed and refined. Many mathematical models, including those used in artificial intelligence, cannot be solved analytically, and to deal with this you will establish and practice fundamental p

Modern artificial intelligence relies on multivariate calculus: every time a neural network learns, it does so by computing derivatives in high-dimensional spaces. Many real-world problems seek to understand the function of a vector, where the vector could be a position in space, a direction, or the weights of a neural network. In this module, you will explore the world of multivariate techniques and multivariate calculus, deepening your understanding of vectors, angles, curves, surfaces and volumes, multidimensional space, and alternative co-ordinate systems. You will encounter multidimensional derivatives, integrals and stationary points, and practice multidimensional analogues of techniqu

Symmetry is central to our understanding of a range of subjects, from the structure of molecules to the roots of polynomials. In this module, you will see how group theory naturally appears whenever we look at symmetry. Using familiar examples, including symmetries of regular polygons, rotations and reflection matrices, roots of 1 in the complex plane, and permutations, you will define what makes a group and how this can provide a unifying language, highlighting connections between seemingly different subjects. You will then transition into mathematical analysis, developing an approach to sequences, limits, and continuity that provides the foundation for calculus. Examining a range of exampl

Never has the collection of data been more widespread than it is now. The extraction of information from massive, often complex and messy, datasets brings many challenges to fields such as statistics, mathematics and computing. Develop the skills and understanding to apply modern statistical and data-science tools to gain insight from contemporary data sets. By addressing challenges from a variety of applications, such as social science, public health, industry and environmental science, you will learn how to perform and present an exploratory data analysis and deploy statistical approaches to analyse data and draw conclusions. You will also develop judgement to critically evaluate the appro

Building on your knowledge of vectors and matrices, this module explores the elegant framework of linear algebra, a powerful mathematical toolkit with remarkably diverse applications across statistical analysis, advanced algebra, graph theory, and machine learning. You'll develop a comprehensive understanding of fundamental concepts, including vector spaces and subspaces, linear maps, linear independence, orthogonality, and the spectral decomposition theorem. Through individual exploration, small-group collaboration, and computational exercises, you'll gain both theoretical insight and practical skills. The module emphasises how these abstract concepts translate into powerful problem-solving

This module provides you with a rigorous understanding of microeconomic principles that underpin sophisticated economic analysis and prepares you for further studies in economics. You will explore key microeconomic concepts including utility maximisation, profit maximisation, cost minimisation, market structures, externalities, information economics, public goods, general equilibrium theory, and welfare economics. To succeed in this module, you will need problem-solving skills and to be proficient in algebra, elementary calculus and logical reasoning.

Statistics allows us to estimate trends and patterns in data and gives a principled way to quantify uncertainty in these estimates. The findings can lead to new insights and support decision-making in fields as diverse as cyber security, human behaviour, finance and economics, medicine, epidemiology, environmental sustainability and many more. Dive into the behaviour of multivariate random variables and asymptotic probability theory, both of which are central to statistical inference. You will then be equipped to explore one of the most fundamental statistical models, the linear regression model, and learn how to apply general statistical inference techniques to multi-parameter statistical m

This module is designed to enhance your strategic thinking skills. You will learn how to use games to model real-world strategic situations, and how to analyse and solve these games in scenarios where players are intelligent and rational. The module covers: normal form games extensive form games Bayesian games games with correlation devices repetitive games behavioural games Additionally, you will have opportunities to play these games with your instructor and classmates. A basic understanding of algebra, calculus and economics is necessary for this module.

In this module you will extend the knowledge of macroeconomics that you developed in your first year. Although the primary focus of the module is on macroeconomic theory, this is taught within the context of current events in the international macroeconomic environment. The topics covered include classical and Keynesian views, unemployment, the government budget constraint, monetary and fiscal policy, intertemporal macroeconomics, economic policy in the open economy, unemployment and inflation, adaptive and rational expectations, policy effectiveness under rational expectations, the economics of independent central banks, and growth theory. To succeed in this module, you will need to apply a

Machine learning is at the heart of modern AI systems, and it is a fundamentally mathematical subject. You will learn this mathematics by discovering how techniques are deployed in several AI systems, including the neural networks that have revolutionised the field. You’ll start by building connections with previously encountered approaches through the unifying concept of a loss function of a parameter vector. For example, with a neural network model the vector input is the set of weights, and the loss function might be the prediction error on a dataset. The goal is to find a vector input that produces a small loss; in the above example, this is known as training the neural net. You will lea

Many of the most important real-world challenges, from predicting climate change, to modelling the spread of disease, are described by equations that cannot be solved analytically. To start, you will be introduced to techniques for tackling such problems, beginning with fundamental numerical methods, such as the trapezium rule and Euler’s method, before progressing to more advanced techniques and quantifying the accuracy, stability and limitations of these methods. Alongside numerical approaches, you will also develop heuristic methods to characterise a system's limiting behaviour.? Other familiar phenomena, such as pulses of light down a fibre optic cable to the shudder of turbulence on a p

You will spend this year working in a graduate-level placement role. This is an ideal opportunity to gain experience in an industry or sector that you might be considering working in once you graduate. Although it's up to you to find your placement we'll support you all the way. Our Careers Service will provide guidance on CVs, applications, interview techniques and creating a digital profile.

Building on the statistical techniques explored so far, you deepen your understanding of both the theoretical underpinnings and practical application of frequentist statistical inference. You will then be introduced to an alternative paradigm: Bayesian statistics. The frequentist perspective views all probabilities in terms of the proportions of outcomes over repeated experimentation and has been the foundation of hypothesis testing and experimental design over years of data-driven science and research. Meanwhile, the increasingly popular Bayesian approach arises directly from Bayes theorem, avoiding hypothetical repeated sampling. As a result, Bayesian statistics is often more intuitive and

This module will introduce you to the field of behavioural and experimental economics. It will equip you with the necessary skills to study how the standard rationality assumptions can be relaxed to account for psychological and cognitive biases as well as social preferences. Additionally, you will be introduced to the tool of experimentation in economics as a means of collecting data to test the various economic theories. Some of the topics covered will include: behavioural finance rational emotions nudging choice under risk time preferences social preferences behavioural game theory

Understanding how data evolves over time is crucial across numerous sectors, from finance and engineering to climate science. Develop the tools to analyse temporal data, detect structural changes and build predictive models. Using changepoint detection algorithms, you will learn how to identify abrupt changes in the mean or variance of a process, or parameters in a regression model. These methods will be introduced from a foundational perspective, developing both computational and mathematical understanding. You will then learn to handle temporal dependence by studying a popular range of time-series models, using these to generate insights about the data and produce forecasts. Throughout the

Models of dynamical systems are fundamental to our understanding of the physical and natural world. Explore a new class of model, the Markov jump process, for the time evolution of dynamical systems such as the evolution of species populations in the wild and the spread of infectious diseases. You will learn how to simulate from these processes and will study methods for understanding their properties and behaviours. Unlike deterministic differential equation models, Markov jump processes are random, allowing for different behaviour every time they are simulated. You will discover how it is often possible to associate a jump process with a related differential equation approximation and that

Data science is transforming the role of information technology in society and in many sectors in which economists work. Machine learning and big data methods have gained popularity as tools in academic, government, industry, and beyond. You will be introduced to big data and machine learning techniques with a focus on economic applications. These techniques are already significantly impacting the field of economics for modelling economic relationship, drawing causal inferences, and making predictions. They will soon become a standard toolbox for economists.

Statistical techniques are often applied to environmental data, such as air temperatures, rainfall or wildfire locations. You will learn about some of the common features of such datasets and how these features are used to design statistical models. You will first be introduced to the Gaussian process model for continuous spatial processes. You will learn about the properties of the Gaussian process and implement this model for spatial data analysis, before investigating methods for point-reference data, such as earthquake or wildfire locations. You will also dip into natural hazard risk management, which seeks to mitigate the effects of events, such as flooding or storms, in a manner that i

This module will introduce you to current economic research. Experts from a variety of fields will teach you in-depth about how economic research is conducted and its real-world applications. By the end of this module, you will be able to: Apply advanced economic methods and concepts to evaluate contemporary economic problems. Critically think about the issues raised by economic research, including their applications and limitations. Present the methods and conclusions of advanced economic research using a wide variety of presentational tools. Consolidate the core knowledge, methods, and analytical skills you have developed throughout your degree. These skills will prepare you for a professi

This module builds on the foundations of monetary and fiscal policy analysis by placing policy decisions in a global context. The first part of the module emphasises the interpretation and analysis of macroeconomic data. You will learn how to apply empirical methods to understand fluctuations in output, employment, inflation, and trade balances. The second part focuses on the design and coordination of monetary and fiscal policy in an interconnected world. Special attention will be given to the challenges that central banks and governments face in managing global shocks. The topics covered will include international policy spillovers, exchange rate regimes, capital flows, and the evolving ro

The study of graphs (mathematical objects used to model networks and pairwise relations between objects) is a cornerstone of discrete mathematics. Graphs can represent important real-world situations, and the study of algorithms for graph-theoretical problems has strong practical significance. You will learn about structural and topological properties of graphs, including graph minors, planarity and colouring. We will introduce several theoretical tools, including matrices relating to graphs and the Tutte polynomial. We will also study fundamental algorithms for network exploration, routing and flows, with applications to the theory of connectivity and trees, considering implementation, proo

This module provides a comprehensive exploration of international trade and global business dynamics, connecting theoretical models with practical policy implications. You will examine core trade theories including the Ricardian model, Heckscher-Ohlin model, and heterogeneous firm models. The module also offers in-depth analyses of international factor mobility, trade policies, and globalisation trends. On the international business side, you will study key topics such as global value chains, multinational firm strategies, international competitive advantage, and the economic impacts of outsourcing and offshoring. The module focuses on real-world applications, exploring how theoretical frame

Mathematical models are central to financial decision making. You will discover the mathematical foundations necessary to model certain transactions in the world of finance. You will then study stochastic models for financial markets and investigate the pricing of European and American options and other financial products. You will explore two discrete models, the binomial model and the finite market model, and one continuous model. Following an introduction to some probabilistic terminology, such as sigma algebras and martingales, and some financial terminology such as arbitrage opportunities and self-financing trading strategies, you will deduce the Black Scholes formula. You will also gai

From denoising diffusion to flow matching, modern generative models are governed by elegant mathematics: stochastic differential equations, PDEs for probability evolution and transport on spaces of measures. This module develops that mathematical toolkit and shows how it underpins today’s state-of-the-art image, audio and scientific generative models.? We start from how probability distributions evolve over time (continuity and Fokker–Planck equations) and show how this leads to a reverse-time stochastic differential equation and an equivalent probability-flow ODE. We then look at discrete-time diffusion models and explain why their training objective is a practical stand-in for maximum like

Public policy analysis is the study of government’s role in the economy. It involves examining both its normative and positive aspects. To gain a comprehensive understanding, we look at a combination of theories, empirical findings, and real-world examples. We begin by focusing on public goods such as water, transportation, and other infrastructure that the government can provide directly or in collaboration with the private sector. This includes looking at the practice of regulators, as well as cost-benefit analysis. We evaluate the trade-off between efficiency and fairness, then examine state financing, including theories of optimal taxation and recent research on tax evasion and avoidance

Statistics and machine learning share the goal of extracting patterns or trends from very large and complex datasets. These patterns are used to forecast or predict future behaviour or interpolate missing information. Learn about the similarities and differences between statistical inference and machine learning algorithms for supervised learning and how the two approaches can be used for classification and prediction. You will explore the class of generalised linear models, which is one of the most frequently used classes of supervised learning model. You will learn how to implement these models, how to interpret their output and how to check whether the model is an accurate representation

Stochastic processes are fundamental to probability theory and statistics and appear in many places in both theory and practice. For example, they are used in finance to model stock prices and interest rates, in biology to model population dynamics and the spread of disease, and in physics to describe the motion of particles. During this module, you will focus on the most basic stochastic processes and how they can be analysed, starting with the simple random walk. Based on a model of how a gambler's fortune changes over time, it questions whether there are betting strategies that gamblers can use to guarantee a win. We will focus on Markov processes, which are natural generalisations of the