Derivatives and Lags

Abstract

STELLA is designed to incrementally add a rate variable to a stock, a process that is called integration. This is very important in finding the solutions to rate or differential equations. However, often we need to do just the opposite, to differentiate or find the slope of a curve or rate of change of a variable. The main reason that we would want to find a rate of change in STELLA is to be able to identify the maximum or minimum value of a variable. We may be generating a stream of numbers that represent the way in which some variable is changing in time, but how do we find the largest or smallest of this stream as these numbers are calculated? We find the derivative of the variable and we keep track of this value. When it goes to zero, the variable has reached a maximum or minimum value. Such an event is used as a trigger to tell the rest of the program to stop changing because the desired condition has been found. For example, we may be modeling a firm that attempts to maximize its profits. We will start the firm off with a low output per time period and calculate the corresponding profit. Then, we incrementally increase output over time, thereby increasing revenues from sale of that output but also production costs. With increasing output, revenues increase steadily but profits increase at a decreasing rate because costs increase, too. At the point at which the profit function reaches a maximum, production should be kept constant so that the firm continues to produce at the maximum profit level.

Honour,’ Tis a derivative from me to mine, And only that I stand for. —William Shakspeare, The Winter’s Tale