Rules of Differentiation Involving Functions of Different Variables 161 Chain Rule 161 Inverse-Function Rule 163 Exercise 7.3 165 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

Fundamental Methods Of Mathematical Economics [PDF

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 34What 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˜ = . 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 †

Fundamental Methods of Mathematical Economics Fourth (PDF) Fundamental Methods of Mathematical Economics Fourth

Limit Theorems 139 Theorems Involving a Single Function 139 Theorems Involving Two Functions 140 Limit of a Polynomial Function 141 Exercise 6.6 141 Solow Growth Model 498 The Framework 498 A Qualitative-Graphic Analysis 500 A Quantitative Illustration 501 Exercise 15.7 502 A Digression on Exponents In discussing polynomial functions, we introduced the term exponents as indicators of the power to which a variable (or number) is to be raised. The expression 62 means that 6 is to be raised to the second power; that is, 6 is to be multiplied by itself, or 62 ≡ 6 × 6 = 36. In general, we define, for a positive integer n, xn ≡ x × x × · · · × x n terms Equilibrium Analysis in Economics The analytical procedure outlined in Chap. 2 will first be applied to what is known as static analysis, or equilibrium analysis. For this purpose, it is imperative first to have a clear understanding of what equilibrium means.The 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. Partial Differentiation 165 Partial Derivatives 165 Techniques of Partial Differentiation 166 Geometric Interpretation of Partial Derivatives 167 Gradient Vector 168 Exercise 7.4 169