First published by MUP in 2001, Pemberton and Rau has become established as the standard textbook on mathematical methods for students of economics. With pre-requisites only of basic algebra and acquaintance with graphs, the book covers all the mathematics needed by students of economics at the undergraduate and master’s level.

*Mathematics for economists* has grown over the years to include the breadth of material needed by today’s students, while retaining the core of calculus, matrix algebra, optimisation and simple dynamics. The second edition added two chapters on dynamic optimisation in discrete and continuous time, emphasising the connection between the two approaches, as well as two chapters on introductory real analysis. The fourth edition added two chapters on probability theory, presented in a way that is most relevant to how probability and expectation are used in economics.

The fifth edition is 80 pages longer than the fourth, with two new chapters and other important new features.

- The first half of the new Chapter 33 on Further Linear Algebra is concerned with vector spaces, orthogonality and projection, leading up to the singular value decomposition and pseudoinverse of a matrix. These concepts are essential for econometrics at the first-year graduate level.
- The second half of Chapter 33 is motivated by the fact that recent work on assignment models and optimal transport theory have made linear programming a more important part of the economist’s toolkit than it has been for many years.
*Mathematics for economists*has always had a section on linear programming in Chapter 2, designed to introduce optimisation and to teach students how to combine algebra with diagrams. The new material goes much more deeply into the theory, emphasising duality and its connections with marginal analysis. - The new Chapter 34, entitled Convex Analysis, returns to the subject of convexity, a favourite topic of earlier chapters, but now with sufficient rigour to make the mathematics used in advanced textbooks on economic theory comprehensible. The last two sections of the chapter contain a detailed application to classical welfare economics and the proof of a version of the Karush-Kuhn-Tucker theorem on non-linear optimisation, which was discussed at length but with less rigour in Chapter 18.
- The new edition follows its predecessors in having both end-of-section exercises and end-of-chapter problems, the latter being more demanding and often drawing on the material of more than one section. The new edition has increased the number of problems from four per chapter to five; the number of exercises has increased substantially. The authors have always favoured active learning and believe that they have finally succeeded in ensuring that every serious point made in the text is backed up by a problem or exercise.
- The new edition contains considerably more material than its predecessors on applications to economics. As is noted in the Preface, this is a book on the mathematical skills economists need, rather than on economics itself, but it has always given applications to numerous branches of economics. The new edition provides a much richer set. There are new problems and exercises about networks, Cournot duopoly, peak load pricing, autoregressive time series, the dynamics of public debt, efficiency wages, least-squares estimation with or without constraints and much else besides.
- As in previous updates, existing material has been improved and clarified. Examples include the interpretation of matrices as linear mappings, the reasons why the rank of a matrix is an interesting number and the model of exhaustible resource extraction in the second chapter on dynamic optimisation.

In sum, the new edition improves, extends and brings up to date a textbook that will be useful to all students, teachers and researchers in economics.

*Mathematics for economists: an introductory textbook *by Malcolm Pemberton and Nicholas Rau is published on 10th October 2023. The paperback edition is £49.99, but **30% off** if you sign up to the MUP newsletter.

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About the authors:

Malcolm Pemberton is Associate Professor of Economics at University College London.

Nicholas Rau is Honorary Senior Lecturer in Economics at University College London.