Consider the following economic example suppose that

Info icon This preview shows pages 3–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Consider the following economic example. Suppose that the Variable Cost of pro- ducing a level of output Q is given by the equation VC = f ( Q ) = Q 2 , where VC is in $ and Q is in kilograms. Then the equation for Marginal Cost (see “The Rules”, in Module 5) is MC = 2 Q , since Average Variable Cost AVC = VC/Q = Q 2 / Q = Q, and MC has the same vertical intercept (=0) as and twice the slope of the AVC function. If we know that the MC function has the linear form MC = 2 Q , we can retrace our steps to arrive at the VC function: VC = Q 2 . This reverse process is essentially what is involved in integration : if MC is the derivative (or slope) of the VC function, then VC is an integral of the MC function. It is not necessarily “ the ” integral of the MC function. What we can- not do is go from the MC curve to the Total Cost (TC) curve, because Total Cost = Variable Cost + Fixed Costs , TC = VC + FC. Since Fixed Costs are constant , they affect Total Cost but not Marginal Cost. Hence, we can get from MC to VC, but then we need MATH MODULE 10: CALCULUS RESULTS FOR THE NON-CALCULUS SPEAKER M10-3 2 1 2 3 4 6 1 1 2 3 4 9 (a) (b) y = 2 x y = x 2 FIGURE M.10-1
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
to add the constant FC (what mathematicians call “the constant of integration”) to VC to get TC. To summarize our sketch so far, if you did some of the exercises in Module 5, you have already been doing calculus, for a special class of functions, namely linear Average and Marginal functions and quadratic Total functions. Table M.10-1 provides some basic formulas for calculating derivatives, which cover virtually all nonlinear and linear functions one can expect to find in an undergraduate economics course. The double- starred (**) questions in the text give you a chance to apply some of these formulas. Partial Derivatives: In both consumption theory and production theory, we make use of partial derivatives. These are discussed in detail in Appendixes 3 and 9, and you should study both Appendixes to get a sense of how they are utilized. The crucial factor to remember, however, is that partial derivatives (in the forms you are likely to encounter them in undergraduate study) have a different notation, but are no more difficult to manipulate, than “regular” derivatives, provided that you treat all variables except the specific independent variable with respect to which you are calculating the (partial) derivative as constants, using the mathematical equivalent of the economist’s ceteris paribus assumption. 1.2 DERIVATIVES, SLOPES, MAXIMA AND MINIMA Economists love maximizing and minimizing, or at least they assume that rational eco- nomic agents do. Economists tend to assume, for instance, that consumers want to max- imize their “utility” or level of satisfaction (subject to a budget constraint), and that M10-4 MATH MODULE 10: CALCULUS RESULTS FOR THE NON-CALCULUS SPEAKER Function: Function: Derivative: Derivative: y = f ( x ) Example dy/dx = y’ = f’ ( x ) Example 1. y = a y = 4 y = 0 y’ = 0 2. y = ax y = 12 x y’ = a y’ = 12 3. y = ax n y = 4 x 3 y’ = nax n – 1 y’ = 12 x 2 y = 3 x 4 y’ = 12 x 3 4. y = f ( x ) + g ( x ) y = 4 x 3 + 3 x y’ = f ’ ( x ) + g’ ( x ) y’ = 12
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern