Abstract
In Chap. 8 we saw that by taking the derivative of the total cost function with respect to the amount of output, we can obtain the marginal cost function. Because an integral is the inverse of the derivative, we may ask if the reverse is true. The answer is, yes, almost. By taking the integral of the marginal cost function we can get the total cost function up to an additive constant (indefinite integral). Whereas the mathematical logic for this will become clear later, the economic logic should be evident to the reader. A knowledge of fixed cost is not contained in the marginal cost function, and, therefore, we will know the total cost function, except for the amount of the fixed cost that we will show by the unknown constant C.
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Notes
- 1.
Or the volume under a surface in case of a double integral and a 4-, 5-, … dimensional space in case of triple, … integrals.
- 2.
After the German mathematician Georg Friedrich Berhard Riemann. There are other types of integrals: the Riemann-Stieltjes integral, which we shall discuss briefly later in this chapter, and the Lebesgue integral, a discussion of which is beyond the scope of the present book. We shall refer to the Riemann integral as the integral.
- 3.
For example, Table of Integrals, Series and Products by Gradshteyn and Ryzhik, translated by Alan Jeffrey, 1980.
- 4.
The Harrod and Domar models are similar in their mechanics, but Harrod’s model involves expectations that are absent from Domar’s model. See “An Essay in Dynamic Theory,” by Roy F. Harrod in Economic Journal (1939) and “Capital Expansion, Rate of Growth and Employment,” by Evsey D. Domar, Econometrica (1946).
- 5.
Strictly speaking, it is the area under a curve with a sign because the area under the x-axis is subtracted from the area above it. Moreover, as we will see later, interchanging the limits of integration, that is, changing the place of x 1 and x 2 in the last expression will change the sign of A.
- 6.
In this definition and in what follows we use X as the random variable and x as its value or realization. For instance, \(P(X \le x) = p\) means that the probability of random variable X being less than or equal to the number x is p.
- 7.
The British mathematician Roger Cotes (1682–1716) was a professor of astronomy in Cambridge and made contributions to the study of logarithms, Newton’s method of interpolation, and numerical integration. He also edited the second edition of Newton’s Principia. Cotes died young and Newton who was 40 years his senior said “if he had lived we might have known something.”
- 8.
Thomas Simpson (1710–1761) worked on many areas of mathematics including calculus, numerical methods, astronomy, and probability theory. He is the author of several high-quality textbooks. Simpson was a self-taught mathematician and for a time he lectured in London coffee houses. This may seem strange today, but coffee houses were known as Penny Universities where customers could listen to lectures on mathematics, art, law, and other subjects while drinking coffee. See A History of the World in Six Glasses by Tom Standage (2005). Simpson’s rule discussed in the text is due to Newton not Simpson. Justice, however, has prevailed as the Newton method discussed in Chap. 10 is due to Simpson.
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Dadkhah, K. (2011). Integration. In: Foundations of Mathematical and Computational Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13748-8_11
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DOI: https://doi.org/10.1007/978-3-642-13748-8_11
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