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 About the Authors Alpha C. Chiang received his Ph.D. from Columbia University in 1954, after earning a B.A. in 1946 from St. John’s University (Shanghai, China) and an M.A. in 1948 from the University of Colorado. In 1954 he joined the faculty of Denison University in Ohio, where he assumed the chairmanship of the Department of Economics in 1961. From 1964 on, he taught at the University of Connecticut where, after 28 years, he became Professor Emeritus of Economics in 1992. He also held visiting professorships at New Asia College of the Chinese University of Hong Kong, Cornell University, Lingnan University in Hong Kong, and Helsinki School of Economics and Business Administration. His publications include another book on mathematical economics: Elements of Dynamic Optimization, Waveland Press, Inc., 1992. Among the honors he received are awards from the Ford Foundation and National Science Foundation fellowships, election to the presidency of the Ohio Association of Economists and Political Scientists, 1963–1964, and listing in Who’s Who in Economics: A Biographical Dictionary of Major Economists 1900–1994, MIT Press. Kevin Wainwright is a faculty member of the British Columbia Institute of Technology in Burnaby, B.C., Canada. Since 2001, he has served as president of the faculty association and program head in the Business Administration program. He did his graduate studies at Simon Fraser University in Burnaby, B.C., Canada, and continues to teach in the Department of Economics there. He specializes in microeconomic theory and mathematical economics.

Fundamental Methods of Mathematical Economics Solution Manual for Fundamental Methods of Mathematical Economics

Ingredients of a Mathematical Model An economic model is merely a theoretical framework, and there is no inherent reason why it must be mathematical. If the model is mathematical, however, it will usually consist of a set of equations designed to describe the structure of the model. By relating a number of variables to one another in certain ways, these equations give mathematical form to the set of analytical assumptions adopted. Then, through application of the relevant mathematical operations to these equations, we may seek to derive a set of conclusions which logically follow from those assumptions. Some Economic Applications of Integrals 464 From a Marginal Function to a Total Function 464 Investment and Capital Formation 465 Present Value of a Cash Flow 468 Present Value of a Perpetual Flow 470 Exercise 14.5 470 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 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 226The Greek Alphabet 655 Mathematical Symbols 656 A Short Reading List 659 Answers to Selected Exercises 662 Index 677 Duality and the Envelope Theorem 435 The Primal Problem 435 The Dual Problem 436 Duality 436 Roy’s Identity 437 Shephard’s Lemma 438 Exercise 13.6 441 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.

Fundamental methods of mathematical economics Fundamental methods of mathematical economics

Access-restricted-item true Addeddate 2019-08-19 14:48:03 Bookplateleaf 0003 Boxid IA1623805 Camera Sony Alpha-A6300 (Control) Collection_set trent External-identifier 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 Differential Equations with a Variable Term 538 Method of Undetermined Coefficients 538 A Modification 539 Exercise 16.6 540 Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis Bangkok Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei TorontoMathematical Economics versus Econometrics The term mathematical economics is sometimes confused with a related term, econometrics. As the “metric” part of the latter term implies, econometrics is concerned mainly with the measurement of economic data. Hence it deals with the study of empirical observations using statistical methods of estimation and hypothesis testing. Mathematical economics, on the other hand, refers to the application of mathematics to the purely theoretical aspects of economic analysis, with little or no concern about such statistical problems as the errors of measurement of the variables under study. In the present volume, we shall confine ourselves to mathematical economics. That is, we shall concentrate on the application of mathematics to deductive reasoning rather than inductive study, and as a result we shall be dealing primarily with theoretical rather than empirical material. This is, of course, solely a matter of choice of the scope of discussion, and it is by no means implied that econometrics is less important. Indeed, empirical studies and theoretical analyses are often complementary and mutually reinforcing. On the one hand, theories must be tested against empirical data for validity before they can be applied with confidence. On the other, statistical work needs economic theory as a guide, in order to determine the most relevant and fruitful direction of research. In one sense, however, mathematical economics may be considered as the more basic of the two: for, to have a meaningful statistical and econometric study, a good theoretical framework—preferably in a mathematical formulation—is indispensable. Hence the subject matter of the present volume should be useful not only for those interested in theoretical economics, but also for those seeking a foundation for the pursuit of econometric studies. Infinite Time Horizon 649 Neoclassical Optimal Growth Model 649 The Current-Value Hamiltonian 651 Constructing a Phase Diagram 652 Analyzing the Phase Diagram 653 Chapter 11 The Case of More than One Choice Variable 291 11.1 The Differential Version of Optimization Conditions 291 First-Order Condition 291 Second-Order Condition 292 Differential Conditions versus Derivative Conditions 293 Nonlinear Difference Equations— The Qualitative-Graphic Approach 562 Phase Diagram 562 Types of Time Path 564 A Market with a Price Ceiling 565 Exercise 17.6 567 Preface This book is written for those students of economics intent on learning the basic mathematical methods that have become indispensable for a proper understanding of the current economic literature. Unfortunately, studying mathematics is, for many, something akin to taking bitter-tasting medicine—absolutely necessary, but extremely unpleasant. Such an attitude, referred to as “math anxiety,” has its roots—we believe—largely in the inauspicious manner in which mathematics is often presented to students. In the belief that conciseness means elegance, explanations offered are frequently too brief for clarity, thus puzzling students and giving them an undeserved sense of intellectual inadequacy. An overly formal style of presentation, when not accompanied by any intuitive illustrations or demonstrations of “relevance,” can impair motivation. An uneven progression in the level of material can make certain mathematical topics appear more difficult than they actually are. Finally, exercise problems that are excessively sophisticated may tend to shatter students’ confidence, rather than stimulate thinking as intended. With that in mind, we have made a serious effort to minimize anxiety-causing features. To the extent possible, patient rather than cryptic explanations are offered. The style is deliberately informal and “reader-friendly.” As a matter of routine, we try to anticipate and answer questions that are likely to arise in the students’ minds as they read. To underscore the relevance of mathematics to economics, we let the analytical needs of economists motivate the study of the related mathematical techniques and then illustrate the latter with appropriate economic models immediately afterward. Also, the mathematical tool kit is built up on a carefully graduated schedule, with the elementary tools serving as stepping stones to the more advanced tools discussed later. Wherever appropriate, graphic illustrations give visual reinforcement to the algebraic results. And we have designed the exercise problems as drills to help solidify grasp and bolster confidence, rather than exact challenges that might unwittingly frustrate and intimidate the novice. In this book, the following major types of economic analysis are covered: statics (equilibrium analysis), comparative statics, optimization problems (as a special type of statics), dynamics, and dynamic optimization. To tackle these, the following mathematical methods are introduced in due course: matrix algebra, differential and integral calculus, differential equations, difference equations, and optimal control theory. Because of the substantial number of illustrative economic models—both macro and micro—appearing here, this book should be useful also to those who are already mathematically trained but still in need of a guide to usher them from the realm of mathematics to the land of economics. For the same reason, the book should not only serve as a text for a course on mathematical methods, but also as supplementary reading in such courses as microeconomic theory, macroeconomic theory, and economic growth and development. We have attempted to retain the principal objectives and style of the previous editions. However, the present edition contains several significant changes. The material on mathematical programming is now presented earlier in a new Chap. 13 entitled “Further Topics in Optimization.” This chapter has two major themes: optimization with inequality constraints and the envelope theorem. Under the first theme, the Kuhn-Tucker conditions are vii

Fundamental Methods of Mathematical Economics - Goodreads

Complex Numbers and Circular Functions 511 Imaginary and Complex Numbers 511 Complex Roots 512 Circular Functions 513 Properties of the Sine and Cosine Functions 515 Euler Relations 517 Alternative Representations of Complex Numbers 519 Exercise 16.2 521 The Constraint Qualification 412 Irregularities at Boundary Points 412 The Constraint Qualification 415 Linear Constraints 416 Exercise 13.2 418

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Suggestions for the Use of This Book Because of the gradual buildup of the mathematical tool kit in the organization of this book, the ideal way of study is to closely follow its specific sequence of presentation. However, some alterations in the sequence of reading is possible: After completing first-order differential equations (Chap. 15) you can proceed directly to optimal control theory (Chap. 20). If going directly from Chap. 15 to Chap. 20, however, the reader may wish to review Sec. 19.5, which deals with two-variable phase diagrams. If comparative statics is not an area of primary concern, you may skip the comparativestatic analysis of general-function models (Chap. 8) and jump from Chap. 7 to Chap. 9. In that case, however, it would become necessary also to omit Sec. 11.7, the comparativestatic portion of Sec. 12.5, as well as the discussion of duality in Chap. 13. Alpha C. Chiang Kevin Wainwright where Q denotes the quantity of output. Since the two equations have different forms, the production condition assumed in each is obviously different from the other. In (2.1), the fixed cost (the value of C when Q = 0) is 75, whereas in (2.2) it is 110. The variation in cost is also different. In (2.1), for each unit increase in Q, there is a constant increase of 10 in C. But in (2.2), as Q increases unit after unit, C will increase by progressively larger amounts. Clearly, it is primarily through the specification of the form of the behavioral equations that we give mathematical expression to the assumptions adopted for a model. As the third type, a conditional equation states a requirement to be satisfied. For example, in a model involving the notion of equilibrium, we must set up an equilibrium condition, which describes the prerequisite for the attainment of equilibrium. Two of the most familiar equilibrium conditions in economics are Dynamics of Market Price 479 The Framework 480 The Time Path 480 The Dynamic Stability of Equilibrium 481 An Alternative Use of the Model 482 Exercise 15.2 483 very good math preparation book for econ major. It could be used to undergraduate text, or graduate review book, since the level is a bit lower. 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