Variables may exist independently, but they do not really become interesting until they are related to one another by equations or by inequalities. At this moment we shall discuss equations only.
A definitional equation sets up an identity between two alternate expressions that have exactly the same meaning. For such an equation, the identical-equality sign = (read: "is identically equal to") is often employed m place of the regular equals sign =f although the latter is also acceptable. As an example, total profit is defined as the excess of total revenue over total cost; we can therefore write n = R-C
A behavioral equation, on the other hand, specifies the manner in which a variable behaves in response to changes in other variables. This may involve either human behavior (such as the aggregate consumption pattern in relation to national income) or nonhuman behavior (such as how total cost of a firm reacts to output changes). Broadly defined, 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
where Q denotes the quantity of output. Since the two equations have different forms, the production condition assumed in caeh 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 QT 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 type7 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 lor the attainment of equilibrium. Two of the most familiar equilibrium conditions in economics are
Qd = Qs [quantity demanded = quantity supplied]
and S = I [intended saving = intended investment]
which pertain, respectively, to the equilibrium of a market model and the equilibrium of the national-income model in its simplest form. Similarly, an optimization model either derives or applies one or more optimization conditions. One such condition that comes easily to mind is the condition
MC = MR [marginal cost — marginal revenue]
in the theory of the firm. Becausc equations of this type are neither definitional nor behavioral, they constitute a class by themselves.
Equations and variables are the essential ingredients of a mathematical model. But since the values that an economic variable takes are usually numerical, a few words should be said about the number system. Here, we shall deal only with so-called real numbers.
Whole numbers such as 1,2.3,... are called positive integers; these are the numbers most frequently used in counting. Their negative counterparts -1, -2, ... are called negative integers; these can be employed, for example, to indicate subzero temperatures (in degrees). The number 0 (zero), on the other hand, is neither positive nor negative, and is in that sense unique. Let us lump all the positive and negative integers and the number zero into a single category, referring to them collectivcly as the set. of all integers.
Integers, ofcourse, do not exhaust all the possible numbers, for we have fractions, such as I,1 and which—if placed on a ruler—would fall between the integers. Also, we have negative fractions, such as — j and — ^.Together, these make up the set of all fractions.
The common property of all fractional numbers is that cadi is expressible as a ratio of two integers. Any number that ean be expressed as a ratio of two integers is eul led a rational number. But integers themselves are also rational, because any integer a can be considered as the ratio n/L 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 dceimal (e.g., ~ — 0.25) or a repeating decimal (e.g., | — 0.3333..,), where some number or series of numbers to the right of the decimal point 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 ex-ample is the number \fl= 1,4142..,, which is a nonrepeating, nonterminahng decimal. Another is the special constant iz = 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"1 numbers, which have to do with the square roots of negative numbers. That concept will be discussed later, in Chap. 16.
2,3 The Concept of Sets _
We have already employed the word set several times. Inasmuch as the eonccpt of sets underlies every branch of modern mathematics, it is desirable to familiarize ourselves at least with its more basic aspects.
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