Mathematical methods for economic theory

Martin J. Osborne

Introduction

This tutorial covers the basic mathematical tools used in economic theory. Knowledge of elementary calculus is assumed; some of the prerequisite material is reviewed in the first section. The main topics are multivariate calculus, concavity and convexity, optimization theory, differential equations, and difference equations. For a complete list, see the table of contents.

The tutorial emphasizes techniques rather than abstract theory. However, the conditions under which each technique is applicable are stated precisely; a guiding principle is “accessible precision”.

I try to include only the minimal assumptions needed for each result, although I deviate from this principle when the weakest possible assumptions are complicated. For each result, I give either a proof or a reference to a proof. Proofs that are visible by default (when a page is loaded) are at the same level as the tutorial; if you understand the results, you should be able to understand those proofs. Proofs that are hidden by default (i.e. are visible only after you click a button) are at a higher level. They are still self-contained, but are mathematically more sophisticated than the material that is visible by default.

All pages are intended to look good on a screen of any size. If a page does not look good on your device, let me know (by click the link at the bottom of any page).

Books

The following four books cover more or less the same material as this tutorial.
  • Michael Carter, Foundations of mathematical economics (MIT Press, 2001)
  • Carl P. Simon and Lawrence Blume, Mathematics for economists (Norton, 1994)
  • Knut Sydsæter, Topics in mathematical analysis for economists (Academic Press, 1981)
  • Knut Sydsæter and Peter J. Hammond, Mathematics for economic analysis (Prentice-Hall, 1995)
The level of Sydsæter and Hammond is approximately the same as the level of this tutorial. Simon and Blume, Sydsæter, and Carter are more advanced.

I have consulted also many more specialized sources, including the following books.

  • Tom M. Apostol, Mathematical analysis, 2nd edition (Pearson, 1974).
  • William E. Boyce and Richard C. DiPrima, Elementary differential equations and boundary value problems, 2nd edition (Wiley, 1969).
  • Earl A. Coddington, An introduction to ordinary differential equations (Prentice-Hall, 1961).
  • Jean Dieudonné, Foundations of modern analysis (Academic Press, 1969).
  • Samuel Goldberg, An introduction to difference equations (Wiley, 1958).
  • George Hadley, Linear algebra (Addison-Wesley, 1961).
  • Geoffrey Heal, Gordon Hughes, and Roger Tarling, Linear algebra and linear economics (Macmillan, 1974).
  • Morris W. Hirsch and Stephen Smale, Differential equations, dynamical systems, and linear algebra (Academic Press, 1974).
  • R. Tyrrell Rockafellar, Convex analysis (Princeton University Press, 1970).
  • Walter Rudin, Principles of mathematical analysis, 2nd edition (McGraw-Hill, 2nd edition 1964, 3rd edition 1976).
  • Michael Spivak, Calculus, 2nd edition (Publish or Perish Press, 1980).
  • Rangarajan K. Sundaram, A first course in optimization theory, (Cambridge University Press, 1996).
  • Akira Takayama, Mathematical economics (1st edition Dryden Press, 1974; 2nd edition Cambridge University Press, 1985).

Notes

  • I welcome comments and suggestions. Please click on the link at the bottom of any page to let me know of errors and confusions.
  • The tutorial is copyrighted. No part of the tutorial may be reproduced or published in any other form without the express prior written permission of Martin J. Osborne. In particular, no part of the tutorial may be posted on any other webpage without the express prior written permission of Martin J. Osborne. (Of course you are very welcome to provide a link to the tutorial from another website.) If you would like to translate the tutorial, please write to me.

Acknowledgments

I am grateful to Kim Border for setting me straight on several points. I have benefitted a lot from his “notes” on various mathematical topics. (They are pitched at a much higher level than this tutorial.) I am grateful also to Omar Sherif Elwakil, who provided detailed comments on the entire tutorial.

I thank also the many readers who have pointed out errors and suggested improvements. Regrettably, until February 2014 I failed to carefully keep track of everyone who has done so. The following list includes everyone who has contributed since February 2014, together with those who contributed earlier and left evidence in my email records: Metin Akyol, Zishan Bhatti, Vadim Borokhov, Charles Bowyer, Bogdan Budescu, Domenico D'Amico, Amal Desai, Devrup Ghatak, Odd Godal, Ujo Goto, Jack Gregory, Tim Heilman, Ibrahim Inal, Ananya Kotia, Nicolas Lepage-Saucier, Alice Lépissier, Francis McDonnell, Wilfred Ngia, Shreya Tayal, Scott Tennican, and Anjalee Sandrasegaran.

I am very grateful to Alejandro Lynch for generous technical advice.