Fundamental Methods of Mathematical Economics (COLLEGE IE (REPRINTS))

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behavioral equations can be used to describe the general institutional setting of a model, including the technological (e.g., production function) and legal (e.g., tax structure) aspects. Before a behavioral equation can be written, however, it is always necessary to adopt definite assumptions regarding the behavior pattern of the variable in question. Consider the two cost functions C = 75 + 10Q (2.1) C = 110 + Q 2 Chiang's Fundamental Methods of Mathematical Economics is an introduction to the mathematics of economics. It starts with a review of algebra and set theory then goes on through calculus, differential equations, matrix algebra, integration. It serves well as a transition from very basic economics up to graduate level economics. Theory behind economic models is discussed and the focus is on mathematical economics, deduction, instead of econometrics and statistical inference or induction.

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The Greek Alphabet 655 Mathematical Symbols 656 A Short Reading List 659 Answers to Selected Exercises 662 Index 677

Step 1: Effect of Nonnegativity Restrictions 403 Step 2: Effect of Inequality Constraints 404 Interpretation of the Kuhn-Tucker Conditions 408 The n-Variable, m-Constraint Case 409 Exercise 13.1 411

Fundamental Methods Of Mathematical Economics [PDF] Fundamental Methods Of Mathematical Economics [PDF]

Total Derivatives 189 Finding the Total Derivative 189 A Variation on the Theme 191 Another Variation on the Theme 192 Some General Remarks 193 Exercise 8.4 193 The common property of all fractional numbers is that each is expressible as a ratio of two integers. Any number that can be expressed as a ratio of two integers is called a rational number. But integers themselves are also rational, because any integer n can be considered as the ratio n/1. The set of all integers and the set of all fractions together form the set of all rational numbers. An alternative defining characteristic of a rational number is that it is expressible as either a terminating decimal (e.g., 14 = 0.25) or a repeating decimal (e.g., 1 = 0.3333 . . .), where some number or series of numbers to the right of the decimal point 3 is repeated indefinitely. Once the notion of rational numbers is used, there naturally arises the concept of irrational numbers—numbers √ that cannot be expressed as ratios of a pair of integers. One example is the number 2 = 1.4142 . . . , which is a nonrepeating, nonterminating decimal. Another is the special constant π = 3.1415 . . . (representing the ratio of the circumference of any circle to its diameter), which is again a nonrepeating, nonterminating decimal, as is characteristic of all irrational numbers. Each irrational number, if placed on a ruler, would fall between two rational numbers, so that, just as the fractions fill in the gaps between the integers on a ruler, the irrational numbers fill in the gaps between rational numbers. The result of this filling-in process is a continuum of numbers, all of which are so-called real numbers. This continuum constitutes the set of all real numbers, which is often denoted by the symbol R. When the set R is displayed on a straight line (an extended ruler), we refer to the line as the real line. In Fig. 2.1 are listed (in the order discussed) all the number sets, arranged in relationship to one another. If we read from bottom to top, however, we find in effect a classificatory scheme in which the set of real numbers is broken down into its component and subcomponent number sets. This figure therefore is a summary of the structure of the real-number system. Real numbers are all we need for the first 15 chapters of this book, but they are not the only numbers used in mathematics. In fact, the reason for the term real is that there are also “imaginary” numbers, which have to do with the square roots of negative numbers. That concept will be discussed later, in Chap. 16.The Concept of Sets We have already employed the word set several times. Inasmuch as the concept of sets underlies every branch of modern mathematics, it is desirable to familiarize ourselves at least with its more basic aspects. Since mathematical economics is merely an approach to economic analysis, it should not and does not fundamentally differ from the nonmathematical approach to economic analysis. The purpose of any theoretical analysis, regardless of the approach, is always to derive a set of conclusions or theorems from a given set of assumptions or postulates via a process of reasoning. The major difference between “mathematical economics” and “literary economics” is twofold: First, in the former, the assumptions and conclusions are stated in mathematical symbols rather than words and in equations rather than sentences. Second, in place of literary logic, use is made of mathematical theorems—of which there exists an abundance to draw upon—in the reasoning process. Inasmuch as symbols and words are really equivalents (witness the fact that symbols are usually defined in words), it matters little which is chosen over the other. But it is perhaps beyond dispute that symbols are more convenient to use in deductive reasoning, and certainly are more conducive to conciseness and preciseness of statement.