Mathematical Methods and Models for Economists

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Preface and Acknowledgments
Much of the time of the average graduate student in economics is spent
learning a new language, that of mathematics. Although the investment does
eventually pay off in many ways, the learning process can be quite painful.
I know because I have been there. I remember the long nights spent puzzling over the mysteries of the Hamiltonian, the frustration of not understanding a single one of the papers in my second macroeconomics reading
list, the culture shock that came with the transition from the undergraduate
textbooks, with their familiar diagrams and intuitive explanations, into
Debreu's Theory of Value, and my despair before the terse and incredibly
dry prose of the mathematics texts where I sought enlightenment about the
arcane properties of contractions.
This book is an attempt to make the transition into graduate economics
somewhat less painful. Although some of my readers may never believe me,
I have tried to do a number of things that should make their lives a bit easier.
The first has been to collect in one place, with a homogeneous notation, most
of the mathematical concepts, results, and techniques that are required to
follow the standard first- and second-year theory courses. I have also tried
to organize this material into a logical sequence and have illustrated its
applications to some of the standard models. And last but not least, I have
attempted to provide rigorous proofs for most of the results as a way to get
the reader used to formal reasoning. Although a lot of effort has gone into
making the text as clear as possible, the result is still far from entertaining.
Most students without a good undergraduate background in mathematics
are likely to find the going a bit rough at times. They have all my sympathy
and the assurance that it does build character.
This book has been long in the making. It started out as a set of notes that
I wrote for myself during my first year at Penn. Those notes were then
refined for the benefit of my younger classmates when I became a teaching
assistant, and they finally grew into lecture notes when I had the misfortune
to graduate and was forced onto the other side of the lectern. Along the way,
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