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

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Note to self: main text for Econ 106: Elements of Mathematical Economics under Prof. Joseph Anthony Y. Lim, First Semester 1996-97, UP School of Economics.

Fundamental Methods of Mathematical Economics - Goodreads

Basic Properties of Determinants 94 Determinantal Criterion for Nonsingularity 96 Rank of a Matrix Redefined 97 Exercise 5.3 98 PART THREE COMPARATIVE-STATIC ANALYSIS 123 Chapter 6 Comparative Statics and the Concept of Derivative 124 6.1 The Nature of Comparative Statics 124 6.2 Rate of Change and the Derivative 125 The Difference Quotient 125 Notes on Vector Operations 59 Multiplication of Vectors 59 Geometric Interpretation of Vector Operations 60 Linear Dependence 62 Vector Space 63 Exercise 4.3 65 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∈SWhat about the complement of a set? To explain this, let us first introduce the concept of the universal set. In a particular context of discussion, if the only numbers used are the set of the first seven positive integers, we may refer to it as the universal set U. Then, with a given set, say, A = {3, 6, 7}, we can define another set A˜ (read: “the complement of A”) as the set that contains all the numbers in the universal set U that are not in the set A. That is, A˜ = {x | x ∈ U The Inflation-Unemployment Model Once More 609 Simultaneous Differential Equations 610 Solution Paths 610 Simultaneous Difference Equations 612 Solution Paths 613 Exercise 19.4 614

Fundamental Methods of Mathematical Economics - McGraw Hill

EXERCISE 3.2 1. Given the market model Qd = Qs Q d = 21 − 3P Q s = −4 + 8P find P ∗ and Q ∗ by (a) elimination of variables and (b) using formulas (3.4) and (3.5). (Use fractions rather than decimals.) 2. Let the demand and supply functions be as follows: (b) Q d = 30 − 2P (a) Q d = 51 − 3P Q s = 6P − 10 Q s = −6 + 5P find P ∗ and Q ∗ by elimination of variables. (Use fractions rather than decimals.) 3. According to (3.5), for Q ∗ to be positive, it is necessary that the expression (ad − bc) have the same algebraic sign as (b + d). Verify that this condition is indeed satisfied in the models of Probs. 1 and 2. 4. If (b + d ) = 0 in the linear market model, can an equilibrium solution be found by using (3.4) and (3.5)? Why or why not? 5. If (b + d ) = 0 in the linear market model, what can you conclude regarding the positions of the demand and supply curves in Fig. 3.1? What can you conclude, then, regarding the equilibrium solution? Again referring to Fig. 2.1, we see that the union of the set of all integers and the set of all fractions is the set of all rational numbers. Similarly, the union of the rational-number set and the irrational-number set yields the set of all real numbers. The Quadratic Formula Equation (3.7) has been solved graphically, but an algebraic method is also available. In general, given a quadratic equation in the form ax 2 + bx + c = 0Functions of Two or More Independent Variables Thus far, we have considered only functions of a single independent variable, y = f (x). But the concept of a function can be readily extended to the case of two or more independent variables. Given a function z = g(x, y) a given pair of x and y values will uniquely determine a value of the dependent variable z. Such a function is exemplified by z = ax + by Laws of Set Operations From Fig. 2.2, it may be noted that the shaded area in diagram a represents not only A ∪ B but also B ∪ A. Analogously, in diagram b the small shaded area is the visual