Optimization of Functions of nVariables

The idea of optimization is fundamental in economies, and the mathematical methods of optimization underlie most economic models. For example, the theory of demand is based on the model of a consumer who chooses the best ( most preferred") bundle of goods from the set of affordable bundles. The theory of supply is based on the model of a lirm choosing inputs in such a way as to minimize the cost of producing any given level of output, and then choosing output to maximize profit. Rationality and optimization are virtually synonymous in economics.

In a formal sense, by optimization we mean the maximization or minimization of a function over some given set. The significance of the concept of optimization is therefore that it gives us a well-defined mathematical procedure for obtaining the solutions to economic models: the "predictions" of the model are based upon the solution to the optimization problem it contains.

We already considered optimization methods for functions of one variable in chapter 6. In this chapter we extend these to functions of any number of variables though, as in chapter 11. we continue to focus on functions that are twice continuously differentiable, namely f e C1. We hegin by considering the unconstrained problem, in which any point in R" is allowed lo be a possible solution. In the last section of this chapter we modify this lo consider the case, often arising in economics. in which permissible values of at least one of the variables are restricted to a subset of the real line. In chapter 13 we consider the important problem of functional constraints.

Few difficulties will be presented by the methods in this chapter and the next if the reader has a good grasp of the material in chapters 6 and 11, and for the discussion of second-order conditions in section 12.2, it will also be useful to have studied the material on quadratic forms in section 10.3 and their relationship to concavity discussed in chapter 11.

12.1 First-Order Conditions

In chapter (i we defined extreme values and stationary values of a function of one variable. Similar definitions apply here. An extreme value is a maximum or a minimum of the function, while a stationary value in the present case has the following definition:

Definition 12.1

A stationary value of a function / over the domain K" occurs a( a point (a* a*)

at which the «-equalities

hold simultaneouslv.

As in the case of functions of one variable, not ail stationary points need give extreme values, because of the possibility of points of inflection. In the case of functions of n variables, there is the further possibility that a stationary value may be a saddle point, at which the function takes on a maximum with respect to changes in some of the x-values and a minimum with respect to others. For example, consider the case of a function of two variables. Holding one variable constant, say a'2, it becomes a function of .V| only. If at some X| =a* we have /i U[, At ) — 0. where a? is the fixed value of .ii, then we know from the discussion in chapter 6 that this could correspond to a minimum, a maximum, or a point of inflection of the function, thought of only as a function of X|. Likewise, if we fix the value of X| and consider the function only as a function of a single variable, x2. the point at which /2(xt.-vi) = 0 could correspond to a minimum, a maximum, or a point of inflection of this function. A further difficulty occurs when considering the possibility of allowing both X| and x2 to change in moving away from the point (,rj\x2). The complications that can arise are similar to those associated with the function J'(x\. x2) = xf + x£ — 5xjx2 discussed in example 11.24 and figure 11.18. This function appears to possess a minimum point at (x*. xi) = (0. 0) as one changes V| only orx2 only, with the function being convex in each of these two directions. However, as we see in figure 11.24. this function does not display a minimum when we change x\ and x2 simultaneously. This means that if we have a point Cx*. x2) at which both partial derivatives are zero

it does not necessarily imply we have either a minimum or a maximum. Figure I2.l illustrates four possibilities. Cases (c) and (d) represent saddle points. Only in cases (a) and (b) do we have extreme values of the function, in the first case a minimum and in the second a maximum.

(at minimum in ¿^-direction, minimum in a,-direction

(b) maximum in a-,-direction, maximum in rn-direction

(at minimum in ¿^-direction, minimum in a,-direction

(b) maximum in a-,-direction, maximum in rn-direction

(ci minimum in a,-direction, maximum (d) maximum in a,-direction, minimum in a,-direction in Aydirection

Figure 12.1 Possible cases of stationary values of f{x\, a-jI

(ci minimum in a,-direction, maximum (d) maximum in a,-direction, minimum in a,-direction in Aydirection

Figure 12.1 Possible cases of stationary values of f{x\, a-jI

The purpose of this discussion is to show that it is not sufficient for (x*_____x," i to yield an extreme value of the function in order that the conditions in definition 12.1 be satisfied. Just as in the case of functions of one variable, we are going to have to develop second-order conditions to help us distinguish among maximum points, minimum points, and other points such as saddle points.

However, we can show that for (a* a*) to yield a maximum or a minimum it is necessary that the conditions in definition 12.1 be satisfied. Consider figure 12.1 (b). If we are at the peak of the hill, it must be impossible to move away from the point (a,', a?) in any direction in such a way as to increase the value of the function. That implies that it must be impossible to move either in the a| -direction or the redirection and increase the value of the function. It is a necessary condition, therefore, that the partial derivatives at this point are zero. A similar argument applies to the case of the minimum in figure 12.1(a). We can put

 Theorem 12.1 If at a point (.**, ..., x*) we have a local maximum of the function /'. so that
0 0