Relative versus Absolute Extremum

If the objective function is a constant Junction, as in Fig. 9. la, all values of the choice variable * will result in the same value of y, and the height of each point on the graph of the function (such as A or B or C) may be considered a maximum or, for that matter, a minimum—or, indeed, neither. In this case, there is in effect no significant choice to be made regarding the value of x for the maximization or minimization ofy.

In Fig. 9.1ft, the function is strictly increasing, and there is no finite maximum if the set of nonnegative real numbers is taken to be its domain, However, we may consider the end point D on the left (the y intercept) as representing a minimum; in fact, it is in this case the absolute (or global) minimum in the range of the function.

The points E and F in Fig. 9,lc? on the other hand, are examples of a relative (or local) extremum, in the sense that each of these points represents an extremum in the immediate neighborhood of the point only. The fact that point F is a relative minimum is, of course, no guarantee that it is also the global minimum of the function, although this may happen to be the case. Similarly, a relative maximum point such as E may or may not be a global maximum. Note also that a function can very well have several relative externa, some of which may be maxima while others are minima.

In most economic problems that we shall be dealing with, our primary, if not exclusive, concern will be with extreme values otheT than end-point values, for with most such problems the domain of the objective function is restricted to be the set of nonnegative real numbers, and thus an end point (oh the left) will represent the zero level of the choice variable, which is often of no practical interest. Actually, the type of function most frequently encountered in economic analysis is that shown in Fig. 9.1c, or some variant thereof that contains only a single bend in the curve. We shall therefore continue our discussion mainly with reference to the search for relative extrema such as points E and F. This will, however, by no means foreclose the knowledge of an absolute maximum if we want it, because an absolute maximum must be either a relative maximum or one of the end points of the

FIGURE 9.1 y


function. Thus if we know all the relative maxima, it is necessary only to select the largest of these and compare it with the end points in order to determine the absolute maximum. The absolute minimum of a function can be found analogously. Hereafter, the extreme values considered will be relative or local ones, unless indicated otherwise.

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