132 we can also distinguish between linear and

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1.3.2 We can also distinguish between linear and nonlinear functions. In Figure M.1-3, (a) M1-4 MATH MODULE 1: FUNCTIONS, GRAPHS, AND THE COORDINATE SYSTEM
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and (c) are linear (or straight-line) functions, while (b) and (d) are nonlinear. Linear func- tions (which are discussed in greater detail in Modules 2 and 3) can be written in a number of different ways, all of which are equivalent to the “slope-intercept” form, y = a + bx , where a is the vertical intercept and b is the slope of the function, and both a and b can be positive, negative, or zero. MATH MODULE 1: FUNCTIONS, GRAPHS, AND THE COORDINATE SYSTEM M1-5 Monotonically increasing function Monotonically decreasing function Constant function Decreasing-and-increasing function (a) (b) (c) (d) y xx y = f ( x ) = ( ) = ( ) = ( ) 00 FIGURE M.1-3
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Nonlinear functions can take on a much wider range of forms. The following are a few examples of nonlinear functions: y = x 2 (M.1.1) •y = | Ï x w | = | x 0.5 | (M.1.2) = 3 x 4 + 14 x 3 – 20 x + 3 [polynomial function] (M.1.3) = log x [logarithmic function] (M.1.4) = e x [exponential function] (M.1.5) Some of these particular functions are discussed more thoroughly in Modules 8 and 9. The main point is that in all of them, the mapping from x to y involves more than sim- ply adding a constant to x and/or multiplying x by a constant factor. In all of them the slope of the function changes as x changes. With so many functions to choose from, why do we so often use linear functions, say for supply and demand functions? The main reason is, purely and simply, mathematical convenience . Assuming that the functions are linear usually simpliFes our calculations. There are two further rationales, however, for assuming linearity. The Frst is that in the neighbourhood of an initial point, a linear function may provide a reasonable approxima- tion of the actual function. The second reason is that many nonlinear functions can be “linearized.” If a constant-elasticity demand curve (see Appendix 4 of the text), for exam- ple, has the form Q D = kP h , then by taking the logarithm of both sides (see Module 8) we can convert it to the form: log Q D = log k + h log P , (M.1.6) which is a linear equation in the logarithms of Q D and P , with log k as the vertical inter- cept and the price-elasticity of demand h as the slope. 1.4 ±UNCTIONS AND ECONOMIC “CAUSALITY” Mathematicians, by convention, typically reserve the horizontal axis of a diagram for the “independent” ( x) variable and the vertical axis of the diagram for the “dependent” ( y) variable. In economists’ graphs, we can distinguish three types of relationship between the horizontal and vertical variables, which are shown in ±igure M.1-4. M1-6 MATH MODULE 1: ±UNCTIONS, GRAPHS, AND THE COORDINATE SYSTEM Y 0 X 0 1 1 0 TR, TC TR 0 TC 0 Q 0 0 (a) D' P 0 0 1 D 1 0 (b) (c) FIGURE M.1-4
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Figure M.1- 4(a) depicts the Total Revenue (TR) and Total Cost (TC) curves (or func- tions) for a perfectly competitive ±rm. Both TR and TC are treated in the diagram as functions of the level of output Q , which is on the horizontal axis.
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