Maclaurin and Taylor Series 242 Maclaurin Series of a Polynomial Function 242 Taylor Series of a Polynomial Function 244 Expansion of an Arbitrary Function 245 Lagrange Form of the Remainder 248 Exercise 9.5 250

Fundamental Methods of Mathematical Economics - Alpha C

Differential Equations with a Variable Term 538 Method of Undetermined Coefficients 538 A Modification 539 Exercise 16.6 540 developed in much the same manner as in the previous edition. However, the topic has been enhanced with several new economic applications, including peak-load pricing and consumer rationing. The second theme is related to the development of the envelope theorem, the maximum-value function, and the notion of duality. By applying the envelope theorem to various economic models, we derive important results such as Roy’s identity, Shephard’s lemma, and Hotelling’s lemma. The second major addition to this edition is a new Chap. 20 on optimal control theory. The purpose of this chapter is to introduce the reader to the basics of optimal control and demonstrate how it may be applied in economics, including examples from natural resource economics and optimal growth theory. The material in this chapter is drawn in great part from the discussion of optimal control theory in Elements of Dynamic Optimization by Alpha C. Chiang (McGraw-Hill 1992, now published by Waveland Press, Inc.), which presents a thorough treatment of both optimal control and its precursor, calculus of variations. Aside from the two new chapters, there are several significant additions and refinements to this edition. In Chap. 3 we have expanded the discussion of solving higher-order polynomial equations by factoring (Sec. 3.3). In Chap. 4, a new section on Markov chains has been added (Sec. 4.7). And, in Chap. 5, we have introduced the checking of the rank of a matrix via an echelon matrix (Sec. 5.1), and the Hawkins-Simon condition in connection with the Leontief input-output model (Sec. 5.7). With respect to economic applications, many new examples have been added and some of the existing applications have been enhanced. A linear version of the IS-LM model has been included in Sec. 5.6, and a more general form of the model in Sec. 8.6 has been expanded to encompass both a closed and open economy, thereby demonstrating a much richer application of comparative statics to general-function models. Other additions include a discussion of expected utility and risk preferences (Sec. 9.3), a profit-maximization model that incorporates the Cobb-Douglas production function (Sec. 11.6), and a two-period intertemporal choice problem (Sec. 12.3). Finally, the exercise problems have been revised and augmented, giving students a greater opportunity to hone their skills. Partial Differentiation 165 Partial Derivatives 165 Techniques of Partial Differentiation 166 Geometric Interpretation of Partial Derivatives 167 Gradient Vector 168 Exercise 7.4 169 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

Mathematical Economics

Leontief Input-Output Models 112 Structure of an Input-Output Model 112 The Open Model 113 A Numerical Example 115 The Existence of Nonnegative Solutions 116 Economic Meaning of the Hawkins-Simon Condition 118 The Closed Model 119 Exercise 5.7 120 It is possible that two given sets happen to be subsets of each other. When this occurs, however, we can be sure that these two sets are equal. To state this formally: we can have S1 ⊂ S2 and S2 ⊂ S1 if and only if S1 = S2 . Note that, whereas the ∈ symbol relates an individual element to a set, the ⊂ symbol relates a subset to a set. As an applica General Market Equilibrium 40 Two-Commodity Market Model 41 Numerical Example 42 n-Commodity Case 43 Solution of a General-Equation System 44 Exercise 3.4 45 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 Commutative, Associative, and Distributive Laws 67 Matrix Addition 67 Matrix Multiplication 68 Exercise 4.4 69

Mathematical economics - Paris School of Economics 1 Mathematical economics - Paris School of Economics

Elegant Yet Lucid Writing Style: Chiang?s strength is the eloquence of the writing and the manner in which it is developed. While the content of the text can be difficult, it is understandable. 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. Chapter 5 Linear Models and Matrix Algebra (Continued) 82 5.1 Conditions for Nonsingularity of a Matrix 82 Necessary versus Sufficient Conditions 82 Conditions for Nonsingularity 84 Rank of a Matrix 85 Exercise 5.1 87Relationships between Sets When two sets are compared with each other, several possible kinds of relationship may be observed. If two sets S1 and S2 happen to contain identical elements, S1 = {2, 7, a, f } 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. 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

Fundamental methods of mathematical economics - Open Library Fundamental methods of mathematical economics - Open Library

Rules of Differentiation Involving Two or More Functions of the Same Variable 152 Sum-Difference Rule 152 Product Rule 155 Finding Marginal-Revenue Function from Average-Revenue Function 156 Quotient Rule 158 Relationship Between Marginal-Cost and Average-Cost Functions 159 Exercise 7.2 160 Samuelson Multiplier-Acceleration Interaction Model 576 The Framework 576 The Solution 577 Convergence versus Divergence 578 A Graphical Summary 580 Exercise 18.2 581About 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. 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

But now, with the Solutions Manual to accompany Fundamental Methods of Mathematical Economics 4th edition you will be able to

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Linearization of a Nonlinear DifferentialEquation System 623 Taylor Expansion and Linearization 624