## Some Concluding Remarks

In the present part of the book, we have covered the basic techniques of optimization. The somewhat arduous journey has taken us (1) from the case of a single choice variable to the more general «-variable case, (2) from the polynomial objective function to the exponential and logarithmic, and (3) from the unconstrained to the constrained variety of extremum.

Most of this discussion consists of the "classical" methods of optimization, with differential calculus as the mainstay, and derivatives of various orders as the primary tools. One weakness of the calculus approach to optimization is its essentially myopic nature. While the first- and second-order conditions in terms of derivatives or differentials can normally locate relative or local extrema without difficulty, additional information or further investigation is often required for identification of absolute or global extrema. Our detailed discussion of concavity, convexity, quasiconcavity, and quasiconvexity is intended as a useful stepping stone from the realm of relative extrema to that of absolute ones.

A more serious limitation of the calculus approach is its inability to cope with constraints in the inequality form. For this reason, the budget constraint in the utility-maximization model, for instance, is stated in the form that the total expenditure be exactly equal to (and not "less than or equal to") a specified sum. In other words, the limitation of the calculus approach makes it necessary to deny the consumer the option of saving part of the available funds. And it is for precisely the same reason that we have not explicitly constrained the choice variables to be nonnegative, as economic common sense may dictate. In fact, we have only been able to use inequalities as model specifications (such as Qa > 0 and Qaa < 0). These play a role in evaluating the signs of mathematical solutions but are not objects of mathematical operations themselves.

We shall deal with the matter of inequality constraints when we study mathematical programming (linear and nonlinear programming), which represents the "nonclassical" approach to optimization. That topic, however, is reserved for Part 6 of the book. Meanwhile, so that you can develop an appreciation of the full sweep of categories of economic analysis—statics —> comparative statics -» dynamics—we shall introduce in Part 5 methods of dynamic analysis. This is also pedagogically preferable, since the mathematical techniques of dynamic analysis are closely related to methods of differential calculus which we have just learned.

For those of you who are anxious to turn to mathematical programming, however, it is perfectly feasible to skip Part 5 and proceed directly to Part 6. No methodological difficulties should arise.

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