微观经济学的数学方法

1、.PAGE ：.；PAGE 258Mathematical methods for economic theory: a tutorialby HYPERLINK chass.utoronto.ca/osborne/index.html t _top Martin J. OsborneTable of contents HYPERLINK chass.utoronto.ca/osborne/MathTutorial/index.html t _top Introduction and instructions 1. HYPERLINK chass.utoronto.ca/osborne/Mat

2、hTutorial/REVF.HTM t _top Review of some basic logic, matrix algebra, and calculus 1.1 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/LOGF.HTM t _top Logic 1.2 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/MATF.HTM t _top Matrices and solutions of systems of simultaneous equations 1.3 HYPERLINK

3、 chass.utoronto.ca/osborne/MathTutorial/IAFF.HTM t _top Intervals and functions 1.4 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/CLCF.HTM t _top Calculus: one variable 1.5 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/CLNF.HTM t _top Calculus: many variables 1.6 HYPERLINK chass.utoronto.ca/os

4、borne/MathTutorial/SKEF.HTM t _top Graphical representation of functions 2. HYPERLINK chass.utoronto.ca/osborne/MathTutorial/CALF.HTM t _top Topics in multivariate calculus 2.1 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/CSIF.HTM t _top Introduction 2.2 HYPERLINK chass.utoronto.ca/osborne/MathT

5、utorial/ECRF.HTM t _top The chain rule 2.3 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/DFIF.HTM t _top Derivatives of functions defined implicitly 2.4 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/DIFF.HTM t _top Differentials and comparative statics 2.5 HYPERLINK chass.utoronto.ca/osborne/M

6、athTutorial/HOMF.HTM t _top Homogeneous functions 3. HYPERLINK chass.utoronto.ca/osborne/MathTutorial/CCVF.HTM t _top Concavity and convexity 3.1 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/CV1F.HTM t _top Concave and convex functions of a single variable 3.2 HYPERLINK chass.utoronto.ca/osborne

7、/MathTutorial/QUFF.HTM t _top Quadratic forms 3.2.1 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/QFDF.HTM t _top Definitions 3.2.2 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/QFFF.HTM t _top Conditions for definiteness 3.2.3 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/QFSF.HTM t _top C

8、onditions for semidefiniteness 3.3 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/CVNF.HTM t _top Concave and convex functions of many variables 3.4 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/QCCF.HTM t _top Quasiconcavity and quasiconvexity 4. HYPERLINK chass.utoronto.ca/osborne/MathTutoria

9、l/MOPF.HTM t _top Optimization 4.1 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/INOF.HTM t _top Introduction 4.2 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/DEFF.HTM t _top Definitions 4.3 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/OPNF.HTM t _top Existence of an optimum 5. HYPERLINK

10、chass.utoronto.ca/osborne/MathTutorial/OPIF.HTM t _top Optimization: interior optima 5.1 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/NENF.HTM t _top Necessary conditions for an interior optimum 5.2 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/LONF.HTM t _top Sufficient conditions for a loca

11、l optimum 5.3 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/GOPF.HTM t _top Conditions under which a stationary point is a global optimum 6. HYPERLINK chass.utoronto.ca/osborne/MathTutorial/MOEF.HTM t _top Optimization: equality constraints 6.1 Two variables, one constraint 6.1.1 HYPERLINK chass.

12、utoronto.ca/osborne/MathTutorial/MENF.HTM t _top Necessary conditions for an optimum 6.1.2 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/ILMF.HTM t _top Interpretation of Lagrange multiplier 6.1.3 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/MELF.HTM t _top Sufficient conditions for a local o

13、ptimum 6.1.4 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/MEGF.HTM t _top Conditions under which a stationary point is a global optimum 6.2 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/MEMF.HTM t _top n variables, m constraints 6.3 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/MEEF.HTM t

14、_top Envelope theorem 7. HYPERLINK chass.utoronto.ca/osborne/MathTutorial/MOIF.HTM t _top Optimization: the Kuhn-Tucker conditions for problems with inequality constraints 7.1 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/KTCF.HTM t _top The Kuhn-Tucker conditions 7.2 HYPERLINK chass.utoronto.ca/

15、osborne/MathTutorial/KTNF.HTM t _top When are the Kuhn-Tucker conditions necessary? 7.3 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/KTSF.HTM t _top When are the Kuhn-Tucker conditions sufficient? 7.4 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/NNCF.HTM t _top Nonnegativity constraints 7.5

16、HYPERLINK chass.utoronto.ca/osborne/MathTutorial/OSMF.HTM t _top Summary of conditions under which first-order conditions are necessary and sufficient 8. HYPERLINK chass.utoronto.ca/osborne/MathTutorial/DEEF.HTM t _top Differential equations 8.1 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/IDEF.

17、HTM t _top Introduction 8.2 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/FDEF.HTM t _top First-order differential equations: existence of a solution 8.3 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/SEPF.HTM t _top Separable first-order differential equations 8.4 HYPERLINK chass.utoronto.ca/o

18、sborne/MathTutorial/FLEF.HTM t _top Linear first-order differential equations 8.5 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/DEQF.HTM t _top Phase diagrams for autonomous equations 8.6 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/SDEF.HTM t _top Second-order differential equations 8.7 HYPE

19、RLINK chass.utoronto.ca/osborne/MathTutorial/SIMF.HTM t _top Systems of first-order linear differential equations 9. HYPERLINK chass.utoronto.ca/osborne/MathTutorial/DIQF.HTM t _top Difference equations 9.1 HYPERLINK chass.utoronto.ca/osborne/MathTutorial/FODF.HTM t _top First-order equations 9.2 HY

20、PERLINK chass.utoronto.ca/osborne/MathTutorial/SODF.HTM t _top Second-order equations Mathematical methods for economic theory: a tutorialby HYPERLINK chass.utoronto.ca/osborne/index.html t _top Martin J. Osborne HYPERLINK chass.utoronto.ca/osborne/MathTutorial/COPYRIGH.HTM Copyright 1997-2003 Marti

21、n J. Osborne. Version: 2003/12/28. 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 HYPERLINK chass.utoronto.ca/osborne/CHARCHEK.HTM t _top character check before using the tutorial

22、. If you system does not pass the test, see the HYPERLINK chass.utoronto.ca/osborne/MathTutorial/TECHTIPS.HTM page of technical information. (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

23、 the symbol font: HYPERLINK chass.utoronto.ca/osborne/MathTutorial/SYMFIX.HTM heres how.) IntroductionThis tutorial is a hypertext version of my lecture notes for a second-year undergraduate course. It covers the basic mathematical tools used in economic theory. Knowledge of elementary calculus is a

24、ssumed; some of the prerequisite material is reviewed in the HYPERLINK chass.utoronto.ca/osborne/MathTutorial/REV.HTM first section. The main topics are multivariate calculus, concavity and convexity, optimization theory, differential equations, and difference equations. The emphasis throughout is o

25、n 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 Mathematics for economic analysis by Knut Sysd

26、aeter 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 more advanced still. The only way to lear

27、n the material is to do the exercises! I welcome comments and suggestions. HYPERLINK mailto:osbornechass.utoronto.ca?subject=Math%20Tutorial%20on%20website Please let me know of errors and confusions. The entire tutorial is HYPERLINK chass.utoronto.ca/osborne/MathTutorial/COPYRIGH.HTM copyrighted, b

28、ut you are welcome to provide a link to the tutorial from your site. (If you would like to translate the tutorial, HYPERLINK mailto:osbornechass.utoronto.ca?subject=Math%20Tutorial%20on%20website please write to me.) Acknowledgments: I have consulted many sources, including the books by Sydsaeter an

29、d 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

30、 and Stephen Smale. I have taken examples and exercises from several of these sources. InstructionsThe tutorial is a collection of main pages, with cross-references to each other, and links to pages of exercises (which in turn have cross-references and links to pages of solutions). The main pages ar

31、e listed in the HYPERLINK chass.utoronto.ca/osborne/MathTutorial/IND.HTM table of contents, which you can go to at any point by pressing the button on the left marked Contents. Each page has navigational buttons on the left-hand side, which you can use to make your way through the main pages. The me

32、aning of each button displays in your browsers status box (at the bottom of the screen for Netscape Navigator) when you put the mouse over that button. On most pages there are ten buttons (though on this initial page there are only six), with the following meanings. Go to the next main page. Go to t

33、he next top-level section. Go back to the previous main page. Go back to the previous top-level section. Go to the main page (text) for this section. Go to the exercises for this section. Go to the solutions to the exercises for this section. Go to the table of contents. Search through all pages of

34、the tutorial for a string. View technical information about viewing and printing pages. If youd like to try using the buttons now, press the black right-pointing arrow (on a yellow background), which will take you to the next main page; to come back here afterwards, press the black left-pointing arr

35、ow on that page. After you follow a link on a main page, press the white Text button to return to the page if you wish to do so before going to the next main page. To help you know where you are, an abbreviated title for the main page to which the buttons on the left correspond is given at the top o

36、f the light yellow panel. (For this page, for example, the abbreviated title is Introduction.) Pages of examples and solutions to exercises have orange backgrounds to make it easier to know where you are. If you get lost, press the Text button or Contents button. TechnicalitiesThe tutorial uses fram

37、es extensively. If your browser doesnt support frames, Im not sure what youll see; I suggest you get a recent version of Netscape Navigator. (Other features that I use may also not be supported by other browsers.) Some very old browsers that support frames do not handle the Back and Forward buttons

38、correctly in frames. HTML has no tags to display math. I have faked the math by using text italic fonts for roman letters, the Windows symbol font for most symbols (gifs for others), small fonts for subscripts and superscripts, and tables for alignments. The result is reasonable using Netscape Navig

39、ator with a 12 or 14 point base font and a relatively high resolution monitor, but may not be so great under other circumstances. If what you see on your screen looks awful, let me know and Ill see if I can do anything about it. MathML, a variant of HTML, has extensive capabilities for beautifully d

40、isplaying math, but is currently supported only by Netscape Navigator 7.1 and its cousins (e.g. Mozilla). I am working on a MathML version of the tutorial. 1. HYPERLINK chass.utoronto.ca/osborne/MathTutorial/REVF.HTM t _top Review of some basic logic, matrix algebra, and calculus1.1 LogicBasicsWhen

41、making precise arguments, we often need to make conditional statements, like if the price of output increases then a competitive firm increases its output or if the demand for a good is a decreasing function of the price of the good and the supply of the good is an increasing function of the price t

42、hen an increase in supply at every price decreases the equilibrium price. These statements are instances of the statement if A then B, where A and B stand for any statements. We alternatively write this general statement as A implies B, or, using a symbol, as A B. Yet two more ways in which we may w

43、rite the same statement are A is a sufficient condition for B, and B is a necessary condition for A. (Note that B comes first in the second of these two statements!) Important note: The statement A B does not make any claim about whether B is true if A is NOT true! It says only that if A is true, th

44、en B is true. While this point may seem obvious, it is sometimes a source of error, partly because we do not always apply the rules of logic in everyday communication. For example, when we say if its fine tomorrow then lets play tennis we probably mean both if its fine tomorrow then lets play tennis

45、 and if its not fine tomorrow then lets not play tennis (and maybe also if its not clear whether the weather is good enough to play tennis tomorrow then Ill call you). When we say if you listen to the radio at 8 oclock then youll know the weather forecast, on the other hand, we do not mean also if y

46、ou dont listen to the radio at 8 oclock then you wont know the weather forecast, because you might listen to the radio at 9 oclock or check on the web, for example. The point is that the rules we use to attach meaning to statements in everyday language are very subtle, while the rules we use in logi

47、cal arguments are absolutely clear: when we make the logical statement if A then B, thats exactly what we meanno more, no less. We may also use the symbol to mean only if or is implied by. Thus B A is equivalent to A B. Finally, the symbol means implies and is implied by, or if and only if. Thus A B

48、 is equivalent to A B and B A. If A is a statement, we write the claim that A is not true as not(A). If A and B are statements, and both are true, we write A and B, and if at least one of them is true we write A or B. Note, in particular, that writing A or B includes the possibility that both statem

49、ents are true. Two rulesRule 1 If the statement A B is true, then so too is the statement (not B) (not A). The first statement says that whenever A is true, B is true. Thus if B is false, A must be falsehence the second statement. Rule 2 The statement not(A and B) is equivalent to the statement (not

50、 A) or (not B). Note the or in the second statement! If it is not the case that both A is true and B is true (the first statement), then either A is not true or B is not true. QuantifiersWe sometimes wish to make a statement that is true for all values of a variable. For example, letting D(p) be the

51、 total demand for tomatoes at the price p, it might be true that D(p) 100 for every price p in the set S. In this statement, for every price is a quantifier. Important note: We may use any symbol for the price in this statement: p is a dummy variable. After having defined D(p) to be the total demand

52、 for tomatoes at the price p, for example, we could write D(z) 100 for every price z in the set S. Given that we just used the notation p for a price, switching to z in this statement is a little odd, BUT there is absolutely nothing wrong with doing so! In this simple example, there is no reason to

53、switch notation, but sometimes in more complicated cases a switch is unavoidable (because of a clash with other notation) or convenient. The point is that in any statement of the form A(x) for every x in the set Y we may legitimately use any symbol instead of x. Another type of statement we sometime

54、s need to make is A(x) for some x in the set Y, or, equivalently, there exists x in the set Y such that A(x). For some x (alternatively there exists x) is another quantifier, like for every x; my comments about notation apply to it. HYPERLINK chass.utoronto.ca/osborne/MathTutorial/LOGX1.HTM Exercise

55、s 1.1 Exercises on logicA, B, and C are statements. The following theorem is true: if A is true and B is not true then C is true.Which of the following statements follow from this theorem? If A is true then C is true. If A is not true and B is true then C is not true. If either A is not true or B is

56、 true (or both) then C is not true. If C is not true then A is not true and B is true. If C is not true then either A is not true or B is true (or both). A and B are statements. The following theorem is true: A is true if and only if B is true.Which of the following statements follow from this theor

57、em? If A is true then B is true. If B is true then A is true. If A is not true then B is not true. If B is not true then A is not true. Let G be a group of people. Assume that for every person A in G, there is a person B in G such that A knows a friend of B. Is it true that for every person B in G,

58、there is a person A in G such that B knows a friend of A? HYPERLINK chass.utoronto.ca/osborne/MathTutorial/LOGX1S.HTM Solutions 1.1 Solutions to exercises on logicOnly (e) follows from the theorem. All four statements follow from the theorem. Yes. In the first statement A and B are variables that ca

59、n stand for any person; they may be replaced by any other two symbolsfor example, A can be replaced by B, and B by A, which gives us the second statement. 1.2 Matrices and solutions of systems of simultaneous equationsMatricesI assume that you are familiar with vectors and matrices and know, in part

60、icular, how to multiply them together. (Do the first few exercises to check your knowledge.) The determinant of the 22 matrix abcdis ad bc. If ad bc 0, the matrix is nonsingular; in this case its inverse is 1 ad bc dbca.(You can check that the product of the matrix and its inverse is the identity ma