Advanced Mathematics for Economists

This book contains a compact, accessible treatment of the main mathematical topics encountered in economics at an advanced level, moving from basic material into the twin areas of static and dynamic optimization.

Nearly half of the book is devoted to a survey of univariate calculus, matrix algebra and multuvariate calculus. This fundamental material is made vigorous by the inclusion of a variety of applications. The later chapters focus on the Lagrange multiplier technique: when it will work, why it works and what economic insights it yields. The properties of maximum value functions and duality are explored, as are the Hamiltonian conditions for dynamic problems in the optimal control format. Dynamic programming and the calculus of variations are also covered.

Much of the discussion proceeds at a heuristic level and by worked example, but the theorems and proofs required by the most analytical user are also to be found. The underlying message is that the language of mathematics can be productive, giving expression to the ideas and facilitating approaches from which insights flow that may be hard to come by in other ways.

The book will be particularly useful for final year undergraduates doing mathematics for economists courses, and postgraduate students.

Preface ix

Glossary of Notation xi

PART I BASIC MATERIAL

1 Univariate Calculus 1

2 Matrix Algebra 39

3 Multivariate Calculus 72

PART II STATIC AND DYNAMIC OPTIMIZATION

4 Equality-Constrained Optimization 101

5 Inequality-Constrained Optimization 115

6 The Maximum Value Function 140

7 Dynamic Optimization 160

Appendix 205

References 211

Answers to exercises 214

Index 227

Peter J. Lambert is the author of Advanced Mathematics for Economists: Static and Dynamic Optimization, published by Wiley. This book contains a compact, accessible treatment of the main mathematical topics encountered in economics at an advanced level, moving from basic material into the twin areas of static and dynamic optimization.

Nearly half of the book is devoted to a survey of univariate calculus, matrix algebra and multuvariate calculus. This fundamental material is made vigorous by the inclusion of a variety of applications. The later chapters focus on the Lagrange multiplier technique: when it will work, why it works and what economic insights it yields. The properties of maximum value functions and duality are explored, as are the Hamiltonian conditions for dynamic problems in the optimal control format. Dynamic programming and the calculus of variations are also covered.

Much of the discussion proceeds at a heuristic level and by worked example, but the theorems and proofs required by the most analytical user are also to be found. The underlying message is that the language of mathematics can be productive, giving expression to the ideas and facilitating approaches from which insights flow that may be hard to come by in other ways.

The book will be particularly useful for final year undergraduates doing mathematics for economists courses, and postgraduate students.