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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
16、. 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 cross-references to each other, an
17、d links to pages of exercises (which in turn have cross-references and links to pages of solutions). The main pages are listed in the 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, whic
18、h you can use to make your way through the main pages. The meaning 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), w
19、ith the following meanings. o Go to the next main page. o Go to the next top-level section. o Go back to the previous main page. o Go back to the previous top-level section. o Go to the main page (text) for this section. o Go to the exercises for this section. o Go to the solutions to the exercises
20、for this section. o Go to the table of contents. o Search through all pages of the tutorial for a string. o 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 th
21、e next main page; to come back here afterwards, press the black left-pointing arrow 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 fo
22、r the main page to which the buttons on the left correspond is given at the top of 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 los
23、t, press the Text button or Contents button. Technicalities The tutorial uses frames 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
24、 very old browsers that support frames do not handle the Back and Forward buttons 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 supe
25、rscripts, and tables for alignments. The result is reasonable using Netscape Navigator 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
26、 about it. MathML, a variant of HTML, has extensive capabilities for beautifully displaying 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. Review of some basic logic, matrix algebra, and calculus1.1
27、 LogicBasicsWhen 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 functi
28、on of the price then 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
29、in which we may write 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
30、 if A is true, then 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
31、 lets play tennis 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 n
32、ot mean also if you 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 rul
33、es we use in logical arguments are absolutely clear: when we make the logical statement if A then B, thats exactly what we mean-no 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
34、 only if. Thus A B 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 possibilit
35、y that both statements 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 false-hence the second statement. Rule 2 The statement not(A and B) is equivalent to
36、 the statement (not 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,
37、letting D(p) be the 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
38、be the total demand 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, the
39、re is no reason to 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 st
40、atement we sometimes 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. Exercises 1.1 Exercises on logic1. A, B, and C a
41、re 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? a. If A is true then C is true. b. If A is not true and B is true then C is not true. c. If either A is not true or B is true (or both) then C is no
42、t true. d. If C is not true then A is not true and B is true. e. If C is not true then either A is not true or B is true (or both). 2. 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 theorem? a. If A is true
43、 then B is true. b. If B is true then A is true. c. If A is not true then B is not true. d. If B is not true then A is not true. 3. 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, ther
44、e is a person A in G such that B knows a friend of A? Solutions 1.1 Solutions to exercises on logic1. Only (e) follows from the theorem. 2. All four statements follow from the theorem. 3. Yes. In the first statement A and B are variables that can stand for any person; they may be replaced by any oth
45、er two symbols-for 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 particular, how to multiply them together. (Do the first f
46、ew 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 d-b-ca.(You can check that the product of the matrix and its inverse is the identity matrix.) The determinant of the 33 matrix abcdefghi
47、is D= a(ei- hf)- b(di- gf)+ c(dh- eg).If D 0 the matrix is nonsingular; in this case its inverse is 1 D D11-D12D13-D21D22-D23D31-D32D33where Dij is the determinant of the 22 matrix obtained by deleting the ith column and jth row of the original 33 matrix. That is, D11= ei- hf, D12= bi- ch, D13= bf- ec, D21= di- fg, D22= ai- cg, D23= af- dc, D31= dh- eg, D
