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Basic Mathematics for Economists - Rosser - Chapter 14

14 Exponential functions, continuous growth and differential equations Use the exponential function and natural logarithms to derive the final sum, initial sum and growth rate when continuous growth takes place. Compare and contrast continuous and discrete growth rates. | 14 Exponential functions continuous growth and differential equations Learning objectives After completing this chapter students should be able to Use the exponential function and natural logarithms to derive the final sum initial sum and growth rate when continuous growth takes place. Compare and contrast continuous and discrete growth rates. Set up and solve linear first-order differential equations. Use differential equation solutions to predict values in basic market and macroeconomic models. Comment on the stability of economic models where growth is continuous. 14.1 Continuous growth and the exponential function In Chapter 7 growth was treated as a process taking place at discrete time intervals. In this chapter we shall analyse growth as a continuous process but it is first necessary to understand the concepts of exponential functions and natural logarithms. The term exponential function is usually used to describe the specific natural exponential function explained below. However it can also be used to describe any function in the format y Ax where A is a constant and A 1 This is known as an exponential function to base A. When x increases in value this function obviously increases in value very rapidly if A is a number substantially greater than 1. On the other hand the value of Ax approaches zero if x takes on larger and larger negative values. For all values of A it can be deduced from the general rules for exponents explained in Chapter 2 that A0 1 and A1 A. Example 14.1 Find the values of y Ax when A is 2 and x takes the following values a 0.5 b 1 c 3 d 10 e 0 f -0.5 g -1 and h -3 1993 2003 Mike Rosser Solution a A0 5 1.41 b A1 2 c A3 8 d A10 1024 e A0 1 f A-0 5 0.71 g A-1 0.50 h A-3 0.13 The natural exponential function In mathematics there is a special number which when used as a base for an exponential function yields several useful results. This number is 2.7182818 to 7 dp and is usually represented by the letter e . You should be able to get this