Direct Restrictions on Variables

In section 6.3 we considered the case in which the variable on which a function was defined was restricted to lie in an interval. For example, if a firm is subject to an output quota, then its output is restricted to lie between zero and some upper limit equal to the quota. In fact in many economic problems it makes sense to restrict the values of the variables to be nonnegative, but it is often implicitly or explicitly assumed that this constraint does not bind at the optimum. Nevertheless, we can often learn interesting things, or resolve puzzles in which the first-order conditions appear to grve strange results, by taking such restrictions explicitly into account.

The results we developed in section 6.3. for functions of one variable extend readily to the case of functions of n variables. Thus suppose that each variable

.r, is resuicted to an interval a, < v, < b, J = I «.It can be the case that for some /.is —oo, and for some (not necessarily the same)/, b, is +oo, but we assume that for at least some /',and/or b, are finite. In what follows our remarks are aimed at these variables.

Suppose that the point x* gives a maximum of the function, subject to the constraint that each ¿¡-value lies in its given interval. For each v, in turn we must then have one of three possible cases, which are illustrated in figure 12.6. (Note that xl, is the vector of fixed values {.v, x*_r jr,V ,,.?*).)

Figure 12.6 Possible solutions when v, must lie in an interval

Case 1 o, < xj < b,. In case I we must have /,<x') = 0. To see (his. consider the component of the total differential df corresponding to Jf,. /,(x*) dx,. If fi(x') ^ 0. then it is possible to find a suitably small dx, with the appropriate sign, such that /¡(x')dx; > 0. This way the function value can be increased, contradicting the fact that it is at a maximum. Thus we must have J] (x*) = 0.

This is, of course, the argument we used for the case in which no constraints were imposed.

Case 2 a, = x *. In case 2 we must have /, (x*) < 0. To sec this, suppose that f,(x' ) > 0. We are free to choose some dx, > 0, since that keeps .v, within the feasible interval, and we then have f, (x*) dx, > 0, contradicting the fact that the function is at a maximum. Thus, we can rule out /¡(x*) > 0 as a possibility. However, if /, (x*) < 0, we could only increase the function value by taking some dxi < 0. which is not permissible because it will violate the constraint. Therefore, we cannot rule out the possibility thu^t ft {x*) < 0, nor indeed that /¡(xm) = 0, since in either case we could not increase the function value by permissible variations in Xf,

Case 3 x* — h,. In case 3 we must have ./¡ (x*) > 0. To see this, suppose that f, (x') < 0. We are free to choose a dx, < 0 such that f, (x*) dx, > 0, and so the function value can be increased without violating the constraint. Therefore we can rule ihis out. On the other hand, if /, (x*) > 0, only a dx-, > 0 could increase the function value, but that violates the constraint, and so the function value cannot be increased. The function value also cannot be increased by small variations in .*, if

We can express these cases more succinctly in

Theorem 12.7

II \* is a solution to the problem

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