The Qualitative-Graphic Approach 495 The Phase Diagram 495 Types of Time Path 496 Exercise 15.6 498

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

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 application of this idea, we may state on the basis of Fig. 2.1 that the set of all integers is a subset of the set of all rational numbers. Similarly, the set of all rational numbers is a subset of the set of all real numbers. How many subsets can be formed from the five elements in the set S = {1, 3, 5, 7, 9}? First of all, each individual element of S can count as a distinct subset of S, such as {1} and {3}. But so can any pair, triple, or quadruple of these elements, such as {1, 3}, {1, 5}, and {3, 7, 9}. Any subset that does not contain all the elements of S is called a proper subset of S. But the set S itself (with all its five elements) can also be considered as one of its own subsets—every element of S is an element of S, and thus the set S itself fulfills the definition of a subset. This is, of course, a limiting case, that from which we get the largest possible subset of S, namely, S itself. At the other extreme, the smallest possible subset of S is a set that contains no element at all. Such a set is called the null set, or empty set, denoted by the symbol or {}. The reason for considering the null set as a subset of S is quite interesting: If the null set is not a / S. But since subset of S ( ⊂ S), then must contain at least one element x such that x ∈ by definition the null set has no element whatsoever, we cannot say that ⊂ S; hence the null set is a subset of S. It is extremely important to distinguish the symbol or {} clearly from the notation {0}; the former is devoid of elements, but the latter does contain an element, zero. The null set is unique; there is only one such set in the whole world, and it is considered a subset of any set that can be conceived. Counting all the subsets of S, including the two limiting cases S and , we find a total of 25 = 32 subsets. In general, if a set has n elements, a total of 2n subsets can be formed from those elements.† † Lccn 83019609 Ocr ABBYY FineReader 11.0 (Extended OCR) Ocr_converted abbyy-to-hocr 1.1.11 Ocr_module_version 0.0.14 Old_pallet IA14991 Openlibrary_edition behavioral equations can be used to describe the general institutional setting of a model, including the technological (e.g., production function) and legal (e.g., tax structure) aspects. Before a behavioral equation can be written, however, it is always necessary to adopt definite assumptions regarding the behavior pattern of the variable in question. Consider the two cost functions C = 75 + 10Q (2.1) C = 110 + Q 2 Derivatives of Exponential and Logarithmic Functions 277 Log-Function Rule 277 Exponential-Function Rule 278 The Rules Generalized 278 The Case of Base b 280 Higher Derivatives 280 An Application 281 Exercise 10.5 282 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

because the n terms in the denominator cancel out n of the m terms in the numerator. Note that the case of x = 0 is ruled out in the statement of this rule. This is because when x = 0, the expression x m /x n would involve division by zero, which is undefined. What if m < n, say, m = 2 and n = 5? In that case we get, according to Rule II, x m−n = x −3 , a negative power of x. What does this mean? The answer is actually supplied by Rule II itself: When m = 2 and n = 5, we have x2 x×x 1 1 = = = 3 x5 x×x×x×x×x x×x×x x

Fundamental Methods of Mathematical Economics - Alpha C Fundamental Methods of Mathematical Economics - Alpha C

Solution by Elimination of Variables One way of finding a solution to an equation system is by successive elimination of variables and equations through substitution. In (3.1), the model contains three equations in three variables. However, in view of the equating of Qd and Qs by the equilibrium condition, we can let Q = Q d = Q s and rewrite the model equivalently as follows: Q = a − bP Q = −c + dPChapter 10 Exponential and Logarithmic Functions 255 10.1 The Nature of Exponential Functions 256 Simple Exponential Function 256 Graphical Form 256 Generalized Exponential Function 257 A Preferred Base 259 Exercise 10.1 260

Fundamental Methods of Mathematical Economics - McGraw Hill

Samuelson Multiplier-Acceleration Interaction Model 576 The Framework 576 The Solution 577 Convergence versus Divergence 578 A Graphical Summary 580 Exercise 18.2 581 Cramer’s Rule 103 Derivation of the Rule 103 Note on Homogeneous-Equation Systems 105 Solution Outcomes for a Linear-Equation System 106 Exercise 5.5 107Chapter 12 Optimization with Equality Constraints 347 12.1 Effects of a Constraint 347 12.2 Finding the Stationary Values 349 Lagrange-Multiplier Method 350 Total-Differential Approach 352 An Interpretation of the Lagrange Multiplier 353 n-Variable and Multiconstraint Cases 354 Exercise 12.2 355 Relationships 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 three types of set operation can be visualized in the three diagrams of Fig. 2.2, known as Venn diagrams. In diagram a, the points in the upper circle form a set A, and the points in the lower circle form a set B. The union of A and B then consists of the shaded area covering both circles. In diagram b are shown the same two sets (circles). Since their intersection should comprise only the points common to both sets, only the (shaded) overlapping portion of the two circles satisfies the definition. In diagram c, let the points in the rectangle be the universal set and let A be the set of points in the circle; then the complement set A˜ will be the (shaded) area outside the circle. 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. representation not only of A ∩ B but also of B ∩ A. When formalized, this result is known as the commutative law (of unions and intersections): A∪B = B∪ A