Mathematics for Economists

Constrained local necessary conditions

Theorem

Assume that we are in the regular case andφk,k= 0 , 1 ,.. .,p are twice

di§erentiable at x 0 ,where x 0 is a local minimum of the constrained

optimization problem then

φ^0 k(x 0 ),h

= 0 , k= 1 , 2 ,.. .,p=)hTL^00 xx(x 0 ,λ)h 0.

For local maximums one has

φ^0 k(x 0 ),h

= 0 , k= 1 , 2 ,.. .,p=)hTL^00 xx(x 0 ,λ)h 0.

Recall that L^00 xx(x 0 ,λ)is the second derivative of the Lagrange function

with respect to x at x 0.