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1、Mathematical methods for economic theory: a tutorialby Martin J. OsborneTable of contents Introduction and instructions 1. Review of some basic logic, matrix algebra, and calculus o 1.1 Logic o 1.2 Matrices and solutions of systems of simultaneous equations o 1.3 Intervals and functions o 1.4 Calcul
2、us: one variable o 1.5 Calculus: many variables o 1.6 Graphical representation of functions 2. Topics in multivariate calculus o 2.1 Introduction o 2.2 The chain rule o 2.3 Derivatives of functions defined implicitly o 2.4 Differentials and comparative statics o 2.5 Homogeneous functions 3. Concavit
3、y and convexity o 3.1 Concave and convex functions of a single variable o 3.2 Quadratic forms 3.2.1 Definitions 3.2.2 Conditions for definiteness 3.2.3 Conditions for semidefiniteness o 3.3 Concave and convex functions of many variables o 3.4 Quasiconcavity and quasiconvexity 4. Optimization o 4.1 I
4、ntroduction o 4.2 Definitions o 4.3 Existence of an optimum 5. Optimization: interior optima o 5.1 Necessary conditions for an interior optimum o 5.2 Sufficient conditions for a local optimum o 5.3 Conditions under which a stationary point is a global optimum 6. Optimization: equality constraints o
5、6.1 Two variables, one constraint 6.1.1 Necessary conditions for an optimum 6.1.2 Interpretation of Lagrange multiplier 6.1.3 Sufficient conditions for a local optimum 6.1.4 Conditions under which a stationary point is a global optimum o 6.2 n variables, m constraints o 6.3 Envelope theorem 7. Optim
6、ization: the Kuhn-Tucker conditions for problems with inequality constraints o 7.1 The Kuhn-Tucker conditions o 7.2 When are the Kuhn-Tucker conditions necessary? o 7.3 When are the Kuhn-Tucker conditions sufficient? o 7.4 Nonnegativity constraints o 7.5 Summary of conditions under which first-order
7、 conditions are necessary and sufficient 8. Differential equations o 8.1 Introduction o 8.2 First-order differential equations: existence of a solution o 8.3 Separable first-order differential equations o 8.4 Linear first-order differential equations o 8.5 Phase diagrams for autonomous equations o 8
8、.6 Second-order differential equations o 8.7 Systems of first-order linear differential equations 9. Difference equations o 9.1 First-order equations o 9.2 Second-order equations Mathematical methods for economic theory: a tutorialby Martin J. OsborneCopyright 1997-2003 Martin J. Osborne. Version: 2
9、003/12/28. THIS TUTORIAL USES CHARACTERS FROM A SYMBOL FONT. If your operating system is not Windows or you think you may have deleted your symbol font, please give your system a character check before using the tutorial. If you system does not pass the test, see the page of technical information. (
10、Note, in particular, that if your browser is Netscape Navigator version 6 or later, or Mozilla, you need to make a small change in the browser setup to access the symbol font: heres how.) IntroductionThis tutorial is a hypertext version of my lecture notes for a second-year undergraduate course. It
11、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 differenc
12、e equations. The emphasis throughout is on techniques rather than abstract theory. However, the conditions under which each technique is applicable are stated precisely. A guiding principle is accessible precision. Several books provide additional examples, discussion, and proofs. The level of Mathe
13、matics for economic analysis by Knut Sysdaeter and Peter J. Hammond (Prentice-Hall, 1995) is roughly the same as that of the tutorial. Mathematics for economists by Carl P. Simon and Lawrence Blume is pitched at a slightly higher level, and Foundations of mathematical economics by Michael Carter is
14、more advanced still. The only way to learn the material is to do the exercises! I welcome comments and suggestions. Please let me know of errors and confusions. The entire tutorial is copyrighted, but you are welcome to provide a link to the tutorial from your site. (If you would like to translate t
15、he tutorial, please write to me.) Acknowledgments: I have consulted many sources, including the books by Sydsaeter and Hammond, Simon and Blume, and Carter mentioned above, Mathematical analysis (2ed) by Tom M. Apostol, Elementary differential equations and boundary value problems (2ed) by William E. Boyce and Richard C. DiPrima, and Differential equations, dynamical systems, and linear algebra by Morris W. Hirsch and Stephen Smale. I have taken examples and exercises from several of these sources. Instructions The tutorial is a collection of main pages, with cro
