I often get asked for math book recommendations for graduate studies and research in Economics. Here are some books that I have found useful myself.
I work in economic theory and macroeconomics and one lesson I learnt the hard way is that it is impossible to be productive in these areas without first acquiring mathematical sophistication at the level of a first-year graduate student in mathematics. This takes time. So the earlier you start, the better.
Areas
Real analysis
Basic
At this level the key things to learn are methods of reading and writing formal proofs and the standard idioms used in constructing epsilon-delta proofs. You should develop a healthy fear of mathematical objects: instead of expecting them to behave according to your intuition you should start expecting them to do the worst possible things allowed by their definitions.
Binmore, Mathematical Analysis: A Straightforward Approach. A gentle introduction by an economist.
Rudin, Principles of Mathematical Analysis. ‘Baby Rudin’ is the ultimate in undergraduate analysis books. I would however not recommend this as a first book since it provides very little in the way of motivation. Read Binmore first. Then read chapters 1–9 of this. The material in the later chapters are better studied from other sources.
Dunham, Calculus Gallery. This book discusses the history of analysis by describing in full mathematical detail some of the contributions of major historical figures from Newton onwards. Read this both to practice your analysis skills and to derive consolation from the fact you are not alone in finding analysis hard. The great masters of the past had to struggle for decades to come up with the right ideas.
Berge, Topological Spaces. Apart from standard material, this book is a very good reference for set-valued functions (correspondences). Every student of economics has come across Berge’s Maximum Theorem. Now learn all about it from the master.
Intermediate
The two most important things to learn at this level are measure theory and the basics of functional analysis. Measure theory is the foundation of modern probability theory which in turn is essential for serious theoretical work in game theory, macroeconomics, econometrics or finance. You will also need measure theory if you want to model economies with continua of agents. (Linear) functional analysis deals with infinite-dimensional vector spaces in an abstract way. This is needed when working with economies with an infinite number of commodities or time periods, as one does in macroeconomics or time-series econometrics.
Folland, Real Analysis: Modern Techniques and Their Applications. The best thing about this book, pitched at the math graduate level, is the selection of topics. You get all the important facts about measure theory, point-set topology and functional analysis in a fast-paced presentation. Also, an introduction to measure-theoretic probability.
Rudin, Real and Complex Analysis. ‘Papa Rudin’ is a classic text. The problems here are much more challenging than those in Folland. The book does not discuss Caratheodory’s Theorem on the extension of measures which is needed for probability theory. There is no systematic treatment of topology — only what is strictly needed for the measure theory is discussed in isolation. The second half of the book on complex analysis is not of direct interest to economists.
Kelley, General Topology. Though first published in 1955, this book still seems to cover, in the words of the author, “What Every Young Analyst Should Know”. It is worth going back to this original for the fast pace and very good exercises.
Measure-theoretic probability
Intermediate
There are excellent texts on probability theory and stochastic processes that don’t require measure theory. Classics are Feller’s An Introduction to Probability Theory and Its Applications, Vol. I and Karlin & Taylor’s A First Course in Stochastic Processes. The series of books by Sheldon Ross — A First Course in Probability, Introduction to Probability Models and Stochastic Processes — are worthy successors.
However as a graduate student in Economics, it is a worthwhile investment to learn probability theory in a measure-theoretic setting. The initial investment in building the measure theoretic appartus make the actual probability theory more elegant and conceptually clear.
Billingsley, Probability and Measure. This is the ultimate reference. If you don’t want to just learn measure-theoretic probability but also want to learn why the theory is set up the way it is, this is the book for you. All the messy technical details are here. For every concept defined Billingsley provides many examples and counterexamples to convince you why the definition is the way it is and how alternative approaches would work.
But this is not a free lunch. First, a lot of material is developed in the exercises, which are quite difficult, though hints are supplied for many of them. Second, to make headway you must have a very good grasp of analysis at at least the ‘Baby Rudin’ level. However, no prior knowledge of measure theory is assumed. Third, in the initial chapters Billingsley interleaves measure theory and probability, first building simple versions of measure-theoretic constructs, using them for a bit, and then building more complicated versions. As a result many ideas get repeated and some of the initial applications cannot use the full power of measure theory. I have not been able to make up my mind on whether this is a good thing to do pedagogically. But in any case don’t expect to reach useful results in a hurry. The Central Limit theorem appears only on page 357.
To capitalize on the popularity of this book Wiley brought out an ‘Anniversary Edition’ in 2012. Unfortunately they chose to re-typeset the book and in the process introduced numerous typos. Hopefully this will get corrected in later editions. Meanwhile the third edition is a safer bet.
Williams, Probability with Martingales. Jacod and Protter, Probability Essentials. These are two short books by eminent probabilists which cover all the important landmarks but at the cost of leaving out much of the intuition-building that you find in Billingsley. Of the two, Williams is the more enlightening.
Dynamics
Basic
Hirsch, Smale and Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos. This book introduces the modern approach to dynamical systems which emphasises qualitative behaviour rather than explicit solutions. The book can be read with only a knowledge of multivariate calculus.
Intermediate
Perko, Differential Equations and Dynamical Systems. More technical details than Hirsch-Smale-Devaney. A clean modern treatment.
Optimization
Basic
Luenberger and Ye, Linear and Nonlinear Programming. Most of this book is about numerical optimization algorithms, but the theoretical chapters provide the best proofs of the Kuhn-Tucker conditions and sensitivity results that I have seen.
Mangasarian, Nonlinear Programming. Theorems of the alternatives deserve to be better known among economics students than they are. Chapter 2 of Managasarian gives the most comprehensive textbook treatment that I know of.
Statistics
Basic
Wasserman, All of Statistics. Econometrics courses spend way too much time on the parametric, frequentist approach to statistics. This books shows you, in not too many pages, how much more there is to the subject.
Rao, Linear Statistical Inference and Its Applications. While broadening our horizons we must not forget our roots. This book is not only teaches you about linear models, it is also an excellent introduction to probability theory, linear algebra and asymptotic statistics.
Intermediate
Schervish, Theory of Statistics. Let me confess that I am unlikely to ever read this book from cover to cover. But this is my go-to reference when I quickly need to look up a definition or result in statistical theory. Covers both frequentist and Bayesian statistics.