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

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Chapter 3 Equilibrium Analysis in Economics 30 3.1 The Meaning of Equilibrium 30 3.2 Partial Market Equilibrium—A Linear Model 31 Constructing the Model 31 Solution by Elimination of Variables 33 Exercise 3.2 34

Fundamental Methods Of Mathematical Economics [PDF

Nonlinear Difference Equations— The Qualitative-Graphic Approach 562 Phase Diagram 562 Types of Time Path 564 A Market with a Price Ceiling 565 Exercise 17.6 567Note 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. Partial Market Equilibrium—A Nonlinear Model 35 Quadratic Equation versus Quadratic Function 35 The Quadratic Formula 36 Another Graphical Solution 37 Higher-Degree Polynomial Equations 38 Exercise 3.3 40 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. Chapter 16 Higher-Order Differential Equations 503 16.1 Second-Order Linear Differential Equations with Constant Coefficients and Constant Term 504 The Particular Integral 504 The Complementary Function 505 The Dynamic Stability of Equilibrium 510 Exercise 16.1 511

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

Higher-Order Linear Differential Equations 540 Finding the Solution 540 Convergence and the Routh Theorem 542 Exercise 16.7 543 The Interaction of Inflation and Unemployment 532 The Phillips Relation 532 The Expectations-Augmented Phillips Relation 533 The Feedback from Inflation to Unemployment 534 The Time Path of π 534 Exercise 16.5 537 Limit Theorems 139 Theorems Involving a Single Function 139 Theorems Involving Two Functions 140 Limit of a Polynomial Function 141 Exercise 6.6 141 and whose magnitudes are accepted as given data only; such variables are called exogenous variables (originating from without). It should be noted that a variable that is endogenous to one model may very well be exogenous to another. In an analysis of the market determination of wheat price (P), for instance, the variable P should definitely be endogenous; but in the framework of a theory of consumer expenditure, P would become instead a datum to the individual consumer, and must therefore be considered exogenous. Variables frequently appear in combination with fixed numbers or constants, such as in the expressions 7P or 0.5R. A constant is a magnitude that does not change and is therefore the antithesis of a variable. When a constant is joined to a variable, it is often referred to as the coefficient of that variable. However, a coefficient may be symbolic rather than numerical. We can, for instance, let the symbol a stand for a given constant and use the expression aP in lieu of 7P in a model, in order to attain a higher level of generality (see Sec. 2.7). This symbol a is a rather peculiar case—it is supposed to represent a given constant, and yet, since we have not assigned to it a specific number, it can take virtually any value. In short, it is a constant that is variable! To identify its special status, we give it the distinctive name parametric constant (or simply parameter). It must be duly emphasized that, although different values can be assigned to a parameter, it is nevertheless to be regarded as a datum in the model. It is for this reason that people sometimes simply say “constant” even when the constant is parametric. In this respect, parameters closely resemble exogenous variables, for both are to be treated as “givens” in a model. This explains why many writers, for simplicity, refer to both collectively with the single designation “parameters.” As a matter of convention, parametric constants are normally represented by the symbols a, b, c, or their counterparts in the Greek alphabet: α, β, and γ . But other symbols naturally are also permissible. As for exogenous variables, in order that they can be visually distinguished from their endogenous cousins, we shall follow the practice of attaching a subscript 0 to the chosen symbol. For example, if P symbolizes price, then P0 signifies an exogenously determined price.

PART 2 Static (or Equilibrium) Analysis

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 Chapter 2 Economic Models 5 2.1 Ingredients of a Mathematical Model 5 Variables, Constants, and Parameters 5 Equations and Identities 6 Solving Simultaneous Dynamic Equations 594 Simultaneous Difference Equations 594 Matrix Notation 596 Simultaneous Differential Equations 599 Further Comments on the Characteristic Equation 601 Exercise 19.2 602

Fundamental methods of mathematical economics - Semantic Scholar

Least-Cost Combination of Inputs 390 First-Order Condition 390 Second-Order Condition 392 The Expansion Path 392 Homothetic Functions 394 Elasticity of Substitution 396 CES Production Function 397 Cobb-Douglas Function as a Special Case of the CES Function 399 Exercise 12.7 401 then S1 and S2 are said to be equal (S1 = S2 ). Note that the order of appearance of the elements in a set is immaterial. Whenever we find even one element to be different in any two sets, however, those two sets are not equal. Another kind of set relationship is that one set may be a subset of another set. If we have two sets S = {1, 3, 5, 7, 9} and T = {3, 7} then T is a subset of S, because every element of T is also an element of S. A more formal statement of this is: T is a subset of S if and only if x ∈ T implies x ∈ S. Using the set inclusion symbols ⊂ (is contained in) and ⊃ (includes), we may then write T ⊂S Xcas as a Programming Environment for Stability Conditions for a Class of Differential Equation Models in Economics It has been 20 years since the last edition of this classic text. Kevin Wainwright, a long time user of the text (British Columbia University and Simon Fraser University), has executed the perfect revision―-he has updated examples, applications and theory without changing the elegant, precise presentation style of Alpha Chiang. Readers will find the wait was worthwhile. Definite Integrals 454 Meaning of Definite Integrals 454 A Definite Integral as an Area under a Curve 455 Some Properties of Definite Integrals 458 Another Look at the Indefinite Integral 460 Exercise 14.3 460

Fundamental methods of mathematical economics

The Real-Number System Equations and variables are the essential ingredients of a mathematical model. But since the values that an economic variable takes are usually numerical, a few words should be said about the number system. Here, we shall deal only with so-called real numbers. Whole numbers such as 1, 2, 3, . . . are called positive integers; these are the numbers most frequently used in counting. Their negative counterparts −1, −2, −3, . . . are called negative integers; these can be employed, for example, to indicate subzero temperatures (in degrees). The number 0 (zero), on the other hand, is neither positive nor negative, and is in that sense unique. Let us lump all the positive and negative integers and the number zero into a single category, referring to them collectively as the set of all integers. Integers, of course, do not exhaust all the possible numbers, for we have fractions, such as 23 , 54 , and 73 , which—if placed on a ruler—would fall between the integers. Also, we have negative fractions, such as − 12 and − 25 . Together, these make up the set of all fractions. Chapter 18 Higher-Order Difference Equations 568 18.1 Second-Order Linear Difference Equations with Constant Coefficients and Constant Term 569 Particular Solution 569 Complementary Function 570 The Convergence of the Time Path 573 Exercise 18.1 575 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. Comparative Statics and the Concept of Derivative 124 Rules of Differentiation and Their Use in Comparative Statics 148 Comparative-Static Analysis of General-Function Models 178

Fundamental methods of mathematical economics Fundamental methods of mathematical economics

Variable Coefficient and Variable Term 483 The Homogeneous Case 484 The Nonhomogeneous Case 485 Exercise 15.3 486 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 The Real-Number System 7 2.3 The Concept of Sets 8 Set Notation 9 Relationships between Sets 9 Operations on Sets 11 Laws of Set Operations 12 Exercise 2.3 14The exercises follows each session is a good examination for the important definitions and theory, clearing any misunderstanding in understand the book, which reveals that the author is really good at teaching. The focus of the exercises is in the basics: understand definitions and correctly use theorems or methods. Nothing advanced or technical there. The reasons I did not give it a five star: the last chapter is so concise and general that I can hardly get anything useful(maybe he really makes efforts in his other book about this topic); you would find it a bit difficult to distinguish vectors and matrices from common letters, since there is no differences in printing in the book(however, the notation in the book really follows common practice which is why I decided to start with this book instead of Simon&Blume); there is no answers for a few chapters. c) AB = ⎡ ⎢⎢ ⎢⎣ (7×12) + (11×3) (7×4) + (11×6) (7×5) + (11×1) (2×12) + (9×3) (2×4) + (9×6) (2×5) + (9×1) (10×12) + (6×3) (10×4) + (6×6) (10×5) + (6×1) ⎤ ⎥⎥ ⎥⎦ = ⎡ ⎢⎢ ⎢⎣ 1179446 51 62 19 1387656 ⎤ ⎥⎥ ⎥⎦=C The Greek Alphabet 655 Mathematical Symbols 656 A Short Reading List 659 Answers to Selected Exercises 662 Index 677 The Nature of Mathematical Economics Mathematical economics is not a distinct branch of economics in the sense that public finance or international trade is. Rather, it is an approach to economic analysis, in which the economist makes use of mathematical symbols in the statement of the problem and also draws upon known mathematical theorems to aid in reasoning. As far as the specific subject matter of analysis goes, it can be micro- or macroeconomic theory, public finance, urban economics, or what not. Using the term mathematical economics in the broadest possible sense, one may very well say that every elementary textbook of economics today exemplifies mathematical economics insofar as geometrical methods are frequently utilized to derive theoretical results. More commonly, however, mathematical economics is reserved to describe cases employing mathematical techniques beyond simple geometry, such as matrix algebra, differential and integral calculus, differential equations, difference equations, etc. It is the purpose of this book to introduce the reader to the most fundamental aspects of these mathematical methods—those encountered daily in the current economic literature.