Exponential and Logarithmic Functions

The Mh-derivative test developed in Chap. 9 equips us for the task of locating the extreme values of any objective function, as long as it involves only one choice variable, possesses derivatives to the desired order, and sooner or later yields a nonzero derivative value at the critical value Jt0. In the examples cited in Chap. 9, however, we made use only of polynomial and rational functions, for which we know how to obtain the necessary derivatives. Suppose that our objective function happened to be an exponential one, such as

Then we are still helpless in applying the derivative criterion, because we have yet to learn how to differentiate such a function. This is what we shall do in the present chaptcr.

Exponential functions, as well as the closely related logarithmic functions, have important applications in economics, especially in connection with growth problems, and in economic dynamics in general. The particular application relevant to the present part of the book, however, involves a class of optimization problems in which the choice variable is time. For example, a certain wine dealer may have a stock of wine, the market value of which is known to increase with time in some prescribed fashion. The problem is to determine the best lime to sell that stock on the basis of the wine-value function, after taking into account the interest cost involved in having the money capital tied up in that stock. Exponential functions may enter into such a problem in two ways. First, the value of the wine may increase with time according to some exponential law of growth. In that event, we would have an exponential wine-value function. Second, when we consider the interest cost* the presence of interest compounding will surely introduce an exponential function into the picture. Thus we must study the nature of exponential functions before we can discuss this type of optimization problem.

Since our primary purpose is to deal with time as a choice variable, let us now switch to the symbol t—in lieu of x—to indicate the independent variable in the subsequent discussion. (However, this same symbol t can very well represent variables other than time also.)

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