in the theory of the firm. Because equations of this type are neither definitional nor behavioral, they constitute a class by themselves. 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. 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.

Fundamental Methods of Mathematical Economics, 3rd Edition Fundamental Methods of Mathematical Economics, 3rd Edition

Given a set with n elements {a, b, c, . . . , n} we may first classify its subsets into two categories: one with the element a in it, and one without. Each of these two can be further classified into two subcategories: one with the element b in it, and one without. Note that by considering the second element b, we double the number of categories in the classification from 2 to 4 (= 22). By the same token, the consideration of the element c will increase the total number of categories to 8 (= 23). When all n elements are considered, the total number of categories will become the total number of subsets, and that number is 2n. Set Notation A set is simply a collection of distinct objects. These objects may be a group of (distinct) numbers, persons, food items, or something else. Thus, all the students enrolled in a particular economics course can be considered a set, just as the three integers 2, 3, and 4 can form a set. The objects in a set are called the elements of the set. There are two alternative ways of writing a set: by enumeration and by description. If we let S represent the set of three numbers 2, 3, and 4, we can write, by enumeration of the elements, S = {2, 3, 4} But if we let I denote the set of all positive integers, enumeration becomes difficult, and we may instead simply describe the elements and write I = {x | x a positive integer} which is read as follows: “I is the set of all (numbers) x, such that x is a positive integer.” Note that a pair of braces is used to enclose the set in either case. In the descriptive approach, a vertical bar (or a colon) is always inserted to separate the general designating symbol for the elements from the description of the elements. As another example, the set of all real numbers greater than 2 but less than 5 (call it J ) can be expressed symbolically as J = {x | 2 < x < 5} Here, even the descriptive statement is symbolically expressed. A set with a finite number of elements, exemplified by the previously given set S, is called a finite set. Set I and set J, each with an infinite number of elements, are, on the other hand, examples of an infinite set. Finite sets are always denumerable (or countable), i.e., their elements can be counted one by one in the sequence 1, 2, 3, . . . . Infinite sets may, however, be either denumerable (set I ), or nondenumerable (set J ). In the latter case, there is no way to associate the elements of the set with the natural counting numbers 1, 2, 3, . . . , and thus the set is not countable. Membership in a set is indicated by the symbol ∈ (a variant of the Greek letter epsilon for “element”), which is read as follows: “is an element of.” Thus, for the two sets S and I defined previously, we may write 2∈S 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 TorontoHigher-Order Linear Differential Equations 540 Finding the Solution 540 Convergence and the Routh Theorem 542 Exercise 16.7 543 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.

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

AB = ⎡ ⎣(4×3) + (7×2) (4×8) + (7×6) (4×5) + (7×7) (9×3) + (1×2) (9×8) + (1×6) (9×5) + (1×7) ⎤ ⎦ = ⎡ ⎣26 74 69 29 78 52 ⎤ ⎦https://www.mediafire.com/file/xvddhpkphp89y2j/Instructor%25E2%2580%2599s_Manual_for_Fundamental_Methods_of_Mathematical_Economics_by_Alpha_C._Chiang%252C_Kevin_Wainwright.pdf/file