Overview of Mathematical Tools for Intermediate Microeconomics
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Overview of Mathematical Tools for Intermediate Microeconomics Christopher Flinn Department of Economics New York University August 2002 Copyright c 2002 Christopher Flinn All° Rights Reserved Contents 1 Functions 2 2 Limits 2 3 Derivatives of a Univariate Function 5 4 Some Rules of Differentiation 9 5 Maximization and Minimization of Univariate Functions 9 6 Multivariate Functions 12 7 Partial Derivatives of Multivariate Functions 14 8TotalDerivatives 17 9 Maximization and Minimization of Multivariate Functions 19 10 Some Important Functions for Economists 23 10.1Linear............................................ 23 10.2 Fixed Coefficients...................................... 23 10.3Cobb-Douglas........................................ 24 10.4Quasi-Linear........................................ 25 1 Economics is a quantitative social science and to appreciate its usefulness in problem solving requires us to make limited use of some results from the differential calculus. These notes are to serve as an overview of definitions and concepts that we will utilize repeatedly during the semester, particularly in the process of solving problems and in the rigorous statements of concepts and definitions. 1Functions Central to virtually all economic arguments is the notion of a function. For example, in the study of consumer choice we typically begin the analysis with the specification of a utility function, from which we later derive a system of demand functions, which can be used in conjunction with the utility function to define an indirect utility function. When studying firm behavior, we work with production functions, from which we define (many types of) cost functions, factor demand functions, firm supply functions, and industry supply functions. You get the idea - functions are with us every step of the way, whether we use calculus or simply graphical arguments in analyzing the problems we confront in economics. Definition 1 A univariate (real) function is a rule associating a unique real number y = f(x) with each element x belongingtoasetX. Thus a function is a rule that associates, for every value x in some set X, a unique outcome y. The particularly important characteristic that we stress is that there is a unique value of y associated with each value of x. There can, in general, be different values of x that yield the same value of y however. When a function also has the property that for every value of y there exists a unique value of x we say that the function is 1-1 or invertible. Example 2 Let y =3+2x. For every value of x there exists a unique value of y, so that this is clearly a function. Note that it is also an invertible function since we can write y =3+2x y 3 x = − . ⇒ 2 Thus there exists a unique value of y for every value of x. Example 3 Let y = x2. As required, for every value of x there exists a unique value of y. However for every positive value of y there exists two possible values of x given by √y and -√y. Hence this is a function, but not an invertible one. 2Limits Let a function f(x) be definedintermsofx. We say that the limit of the function as x becomes arbitrarily close to some value x0 is A, or lim f(x)=A. x x0 → 2 Technically this means that for any ε > 0 there exists a δ such that x x0 < δ f(x) A < ε, | − | ⇒ | − | where z denotes the absolute value of z (i.e. z = z if z 0 and z = z if z<0). | | | | ≥ | | − The limit of a function at a point x0 need not be equal to the value of the function at that point, that is lim f(x)=A x x0 → does not necessarily imply that f(x0)=A. In fact, it may be that f(x0) is not even defined. Example 4 Let y = f(x)=3+2x. Then as x 3,y 9, or the limit of f(x)=9as x 3. This is shown by demonstrating that f(x) 9 < ε for→ any →ε if x 3 is sufficiently small. Note→ that | − | | − | f(x) 9 = 3+2x 9 | − | | − | = 2x 6 =2x 3 . | − | | − | If we choose δ = ε/2, then if x 3 < δ, | − | 2 x 3 = 2x 6 = f(x) 9 < 2δ = ε. | − | | − | | − | Thus for any ε, no matter how small, by setting δ = ε/2, x 3 < δ implies that f(x) 9 < ε. Then by definition | − | | − | lim f(x)=9. x 3 → It so happens that in this case f(3) = 9, but these are not the same thing. Definition 5 When limx x0 f(x)=f(x0) we say that the function is continuous at x0. → Then our example f(x)=3+2x is continuous at the value x0 =3. You should be able to convince yourself that this function is continuous everywhere on the real line R ( , ). An example of a function that is not continuous everywhere on R is the following. ≡ −∞ ∞ Example 6 Let the function f be defined as x2 if x =0 f(x)= 6 . ( 1 if x =0 First, consider the limit of f(x) for any point not equal to 0, say for example at x =2. Now f(2) = 4. For any ε > 0, f(x) 4 < ε is equivalent to x2 4 < ε, or x 2 x +2 < ε. At points near x =2, x +2 4.| It is− clearly| the case that x +2| −<|5, say, for| −x su|| fficiently| close to 2. Then let δ =| ε/5. Then| → x 2 < δ implies that ε >| 5 x | 2 > x +2 x 2 = f(x) 4 . Thus | − | | − | | || − | | − | there exists a δ (in this case we have chosen δ = ε/5) such that if x is within δ of x0 =2(that is, x 2 < δ), then f(x) is within ε of the value 4, and this holds for any choice of ε. We also note that| − since| f(2) = 4, the function is continuous at 2. 3 We note that the function is continuous in this example is continuous at all points other than x0 =0. To see this, note the following. Example 7 (Continued) The limit of f(x) at 0 is given by lim f(x)=0. x 0 → This is the case because for any arbitrary ε > 0 there exists a δ > 0 such that x < δ implies that x2 0 < ε. However, since f(0) = 1, we have | | | − | lim f(x) = f(0), x 0 → 6 so that the function is not continuous at the point x0 =0. As we remarked above, it is continuous everywhere else on R. On an intuitive level, we say that a function is continuous if the function can be drawn without ever lifting one’s pencil from the sheet of paper. In Example 4 the function was a straight line, and this function obviously can be drawn without ever lifting one’s pencil (although it would take a long time to draw the line since it extends indefinitely in either direction). On the other hand, in Example 6 we have to lift our pencil at the value x =0, wherewehavetomoveuptoplotthe point f(0) = 1. Thus this function is not continuous at that one point. To rigorously define continuity, and, in the next section, differentiability, it will be useful to introduce the concepts of the left hand limit and the right hand limit. These are only distinguished by the fact that the left hand limit of the function at some point x0 is defined by taking increasing large values of x that become arbitrarily close to x0, while the right hand limit of the function is obtained by taking increasingly smaller values of x that become arbitrarily close to the point x0. Up to this point we have been describing the limiting operation implicitly in terms of the right hand limit. Now let’s explicitly distinguish between the two. Definition 8 Let ∆x>0. At a point x0 the left hand limit of the function f is defined as L1(x0)= lim f(x0 ∆x), ∆x 0 − → and the right hand limit is defined as R1(x0)= lim f(x0 + ∆x). ∆x 0 → We then have the following result. Definition 9 Afunctionf is continuous at the point x0 if and only if L1(x0)=R1(x0)=f(x0). We say that the function is continuous everywhere on its domain if L1(x)=R1(x) for all x in the domain of f. 4 3 Derivatives of a Univariate Function Armedwiththedefinition of a function, continuity, and limits, we have all the ingredients to define the derivatives of a univariate function. A derivative is essentially just the rate of a change of a function evaluated at some point. The section heading speaks of derivatives in the plural since we can speak of the rate of change of the function itself (first derivative), the rate of change of the rate of change of the function (the second derivative), etc. Don’t become worried, we shall never need to use anything more than the second derivative in this course (and that rarely). Consider a function f(x) which is continuous at the the point x0. We can of course “perturb” the value of x0 by a small amount, let’s say by ∆x. ∆x could be positive or negative, but without any loss of generality let’s say that it is positive. Thus the “new” value of x is x0 plus the change, or x0 + ∆x. Now we can consider how much the function changes when we move from the point x0 to the point x0 + ∆x. The function change is given by ∆y = f(x0 + ∆x) f(x0). − Thus the rate of change (or average change) in the function is given by ∆y f(x0 + ∆x) f(x0) = − .