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

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Digression on Inequalities and Absolute Values 136 Rules of Inequalities 136 Absolute Values and Inequalities 137 Solution of an Inequality 138 Exercise 6.5 139 Application to Market and National-Income Models 107 Market Model 107 National-Income Model 108 IS-LM Model: Closed Economy 109 Matrix Algebra versus Elimination of Variables 111 Exercise 5.6 111 What is the complement of U? Since every object (number) under consideration is included in the universal set, the complement of U must be empty. Thus U˜ = .

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

A book should not be rated simply according to its level. Thus, though it is a easy cake, I would recommend it to anyone wishing to have a concrete math foundation for further econ study. It really use econ theories, especially econ models to explain how to use the methods or theory. In fact, you could almost see all the major models in both Micro and Macro. Equations and Identities Variables may exist independently, but they do not really become interesting until they are related to one another by equations or by inequalities. At this moment we shall discuss equations only. In economic applications we may distinguish between three types of equation: definitional equations, behavioral equations, and conditional equations. A definitional equation sets up an identity between two alternate expressions that have exactly the same meaning. For such an equation, the identical-equality sign ≡ (read: “is identically equal to”) is often employed in place of the regular equals sign =, although the latter is also acceptable. As an example, total profit is defined as the excess of total revenue over total cost; we can therefore write π ≡ R−C A behavioral equation, on the other hand, specifies the manner in which a variable behaves in response to changes in other variables. This may involve either human behavior (such as the aggregate consumption pattern in relation to national income) or nonhuman behavior (such as how total cost of a firm reacts to output changes). Broadly defined, Nonalgebraic Functions Any function expressed in terms of polynomials and/or roots (such as square root) of polynomials is an algebraic function. Accordingly, the functions discussed thus far are all algebraic. However, exponential functions such as y = b x , in which the independent variable appears in the exponent, are nonalgebraic. The closely related logarithmic functions, such as y = logb x, are also nonalgebraic. These two types of function have a special role to play in certain types of economic applications, and it is pedagogically desirable to postpone their discussion to Chap. 10. Here, we simply preview their general graphic shapes in Fig. 2.8e and f. Other types of nonalgebraic function are the trigonometric (or circular) functions, which we shall discuss in Chap. 16 in connection with dynamic analysis. We should add here that nonalgebraic functions are also known by the more esoteric name of transcendental functions. Published by McGraw-Hill, an imprint of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, Set Notation A set is simply a collection of distinct objects. These objects may be a group of (distinct) numbers, persons, food items, or something else. Thus, all the students enrolled in a particular economics course can be considered a set, just as the three integers 2, 3, and 4 can form a set. The objects in a set are called the elements of the set. There are two alternative ways of writing a set: by enumeration and by description. If we let S represent the set of three numbers 2, 3, and 4, we can write, by enumeration of the elements, S = {2, 3, 4} But if we let I denote the set of all positive integers, enumeration becomes difficult, and we may instead simply describe the elements and write I = {x | x a positive integer} which is read as follows: “I is the set of all (numbers) x, such that x is a positive integer.” Note that a pair of braces is used to enclose the set in either case. In the descriptive approach, a vertical bar (or a colon) is always inserted to separate the general designating symbol for the elements from the description of the elements. As another example, the set of all real numbers greater than 2 but less than 5 (call it J ) can be expressed symbolically as J = {x | 2 < x < 5} Here, even the descriptive statement is symbolically expressed. A set with a finite number of elements, exemplified by the previously given set S, is called a finite set. Set I and set J, each with an infinite number of elements, are, on the other hand, examples of an infinite set. Finite sets are always denumerable (or countable), i.e., their elements can be counted one by one in the sequence 1, 2, 3, . . . . Infinite sets may, however, be either denumerable (set I ), or nondenumerable (set J ). In the latter case, there is no way to associate the elements of the set with the natural counting numbers 1, 2, 3, . . . , and thus the set is not countable. Membership in a set is indicated by the symbol ∈ (a variant of the Greek letter epsilon for “element”), which is read as follows: “is an element of.” Thus, for the two sets S and I defined previously, we may write 2∈SIt is obvious that intersection is a more restrictive concept than union. In the former, only the elements common to A and B are acceptable, whereas in the latter, membership in either A or B is sufficient to establish membership in the union set. The operator symbols √ ∩ and ∪—which, incidentally, have the same kind of general status as the symbols , +, ÷, etc.—therefore have the connotations “and” and “or,” respectively. This point can be better appreciated by comparing the following formal definitions of intersection and union: Intersection: which pertain, respectively, to the equilibrium of a market model and the equilibrium of the national-income model in its simplest form. Similarly, an optimization model either derives or applies one or more optimization conditions. One such condition that comes easily to mind is the condition MC = MR Operations on Sets When we add, subtract, multiply, divide, or take the square root of some numbers, we are performing mathematical operations. Although sets are different from numbers, one can similarly perform certain mathematical operations on them. Three principal operations to be discussed here involve the union, intersection, and complement of sets. To take the union of two sets A and B means to form a new set containing those elements (and only those elements) belonging to A, or to B, or to both A and B. The union set is symbolized by A ∪ B (read: “A union B”). Lccn 83019609 Ocr ABBYY FineReader 11.0 (Extended OCR) Ocr_converted abbyy-to-hocr 1.1.11 Ocr_module_version 0.0.14 Old_pallet IA14991 Openlibrary_edition Logarithmic Functions 272 Log Functions and Exponential Functions 272 The Graphical Form 273 Base Conversion 274 Exercise 10.4 276

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which is again an expression in terms of parameters only. Since the denominator (b + d) is positive, the positivity of Q ∗ requires that the numerator (ad − bc) be positive as well. Hence, to be economically meaningful, the present model should contain the additional restriction that ad > bc. The meaning of this restriction can be seen in Fig. 3.1. It is well known that the P ∗ and Q ∗ of a market model may be determined graphically at the intersection of the demand and supply curves. To have Q ∗ > 0 is to require the intersection point to be located above the horizontal axis in Fig. 3.1, which in turn requires the slopes and vertical intercepts of the two curves to fulfill a certain restriction on their relative magnitudes. That restriction, according to (3.5), is ad > bc, given that both b and d are positive. The intersection of the demand and supply curves in Fig. 3.1, incidentally, is in concept no different from the intersection shown in the Venn diagram of Fig. 2.2b. There is one difference only: Instead of the points lying within two circles, the present case involves the points that lie on two lines. Let the set of points on the demand and supply curves be denoted, respectively, by D and S. Then, by utilizing the symbol Q (= Q d = Q s ), the two sets and their intersection can be written D = {( P, Q) | Q = a − bP} S = {( P, Q) | Q = −c + dP} and Second-Derivative Test 233 Necessary versus Sufficient Conditions 234 Conditions for Profit Maximization 235 Coefficients of a Cubic Total-Cost Function 238 Upward-Sloping Marginal-Revenue Curve 240 Exercise 9.4 241 where the + part of the ± sign yields x1∗ and the − part yields x2∗ . Also note that as long as b2 − 4ac > 0, the values of x1∗ and x2∗ would differ, giving us two distinct real numbers as the roots. But in the special case where b2 − 4ac = 0, we PART FOUR OPTIMIZATION PROBLEMS 219 Chapter 9 Optimization: A Special Variety of Equilibrium Analysis 220 9.1 Optimum Values and Extreme Values 221 9.2 Relative Maximum and Minimum: First-Derivative Test 222 Relative versus Absolute Extremum 222 First-Derivative Test 223 Exercise 9.2 226 in the theory of the firm. Because equations of this type are neither definitional nor behavioral, they constitute a class by themselves.

and as a special case, we note that x 1 = x . From the general definition, it follows that for positive integers m and n, exponents obey the following rules: (for example, x 3 × x 4 = x 7 ) The total cost C of a firm per day is a function of its daily output Q: C = 150 + 7Q. The firm has a capacity limit of 100 units of output per day. What are the domain and the range of the cost function? Inasmuch as Q can vary only between 0 and 100, the domain is the set of values 0 ≤ Q ≤ 100; or more formally, Domain = {Q | 0 ≤ Q ≤ 100} As for the range, since the function plots as a straight line, with the minimum C value at 150 (when Q = 0) and the maximum C value at 850 (when Q = 100), we have Range = {C | 150 ≤ C ≤ 850} Beware, however, that the extreme values of the range may not always occur where the extreme values of the domain are attained. Equation (3.3) can be viewed as the result of setting the linear function (b + d ) P − (a + c) equal to zero.

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I am only interested in empirical research, using statistical techniques, and I was looking for a good book that could prepare me for studies in mathematical statistics, this book covered more than I needed. Chapter 20 Optimal Control Theory 631 20.1 The Nature of Optimal Control 631 Illustration: A Simple Macroeconomic Model 632 Pontryagin’s Maximum Principle 633 This is a quadratic equation because the left-hand expression is a quadratic function of variable P. A major difference between a quadratic equation and a linear one is that, in general, the former will yield two solution values. For mathematical economists this book covers the minimum amount of mathematics that you need, after this book you can branch out into other specializations such as advanced calculus and real analysis.DYNAMIC ANALYSIS 443 Chapter 14 Economic Dynamics and Integral Calculus 444 14.1 Dynamics and Integration 444 14.2 Indefinite Integrals 446 The Nature of Integrals 446 Basic Rules of Integration 447 Rules of Operation 448 Rules Involving Substitution 451 Exercise 14.2 453 in which y is expressed as a ratio of two polynomials in the variable x, is known as a rational function. According to this definition, any polynomial function must itself be a rational function, because it can always be expressed as a ratio to 1, and 1 is a constant function. A special rational function that has interesting applications in economics is the function a y= or xy = a x which plots as a rectangular hyperbola, as in Fig. 2.8d. Since the product of the two variables is always a fixed constant in this case, this function may be used to represent that special demand curve—with price P and quantity Q on the two axes—for which the total † Autonomous Problems 644 20.4 Economic Applications 645 Lifetime Utility Maximization 645 Exhaustible Resource 647 Exercise 20.4 649 Commutative, Associative, and Distributive Laws 67 Matrix Addition 67 Matrix Multiplication 68 Exercise 4.4 69