The Bordered Hessian In Constrained Optlmization

12.19. Maximize utility u = 2xy subject to a budget constraint equal to 3x + Ay =* 90 by (a) finding the critical values x. y, and  and (b) using the bordered Hessian |H| to test the second-order condition.

a) I he Lagrangian function ts U « 2xy + A(90 - 3x - 4y)

The first-order conditions arc

U, = 2y - 3A =» 0 U, = It - 4A = 0 (/, = 90 - ix - 4y - 0

In matrix form.

Solving by Cramer's rule. ¡A - 48. A,| = 720. | A,| - 540. and | A,i - 360. Thus, x - 15. y - 11.25. and  - 7.5.

bI taking (he second partial* of I with respect to i and y and the rirst pjtlials ol the constraint with respcct ro i and v to form the bordered llevuan.

From Section I J.*

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