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gy

Zrx

The determinant to the right, often referred to as a bordered HessiaD, shall be denoted by \H |, where the bar on top symbolizes the border. On the basis of . this, we may conclude that, given a stationary value of z = f(x, y) or of Z = f(x, y) + X[c - g(x, >■)], a positive \H | is sufficient to establish it as a relative maximum of z; similarly, a negative \H \ is sufficient to establish it as a minimum—all the derivatives involved in \H | being evaluated at the critical values of_x and y.

Now that we have derived the second-order sufficient condition, it is an easy matter to verify that, as earlier claimed, the satisfaction of this condition will guarantee that the endogenous-variable Jacobian (12.12) does not vanish in the optimal state. Substituting (12.18) into (12.12), and multiplying both the first column and the first row of the Jacobian by — 1 (which will leave the value of the determinant unaltered), we see that

0 8v

That is. the endogenous-variable Jacobian is identical with the bordered Hessian — a result similar to (11.42) where it was shown that, in the free-extremum context, the endogenous-variable Jacobian is identical with the plain Hessian. If, in fulfillment of the sufficient condition, we have | H | =h 0 at the optimum, then |/| must also be nonzero. Consequently, in applying the implicit-function theorem to the present context, it would not be amiss to substitute the condition \H | + 0 for the usual condition \J\ =*= 0. This practice will be followed when we analyze the comparative statics of constrained-optimization problems below.

Example 2 Let us now return to Example 1 of Sec. 12.2 and ascertain whether the stationary value found there gives a maximum or a minimum. Since Zx = y — A and Zv = x — X. the second-order partial derivatives are Zvv = 0, Zxy = Zyx = 1, and Zvl = 0. The border elements we need are gx = 1 and gy = 1. Thus we find that

0 0

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