Mathematics for Economists

Constrained local su¢ cient conditions

One must show that

φ 0 (x)K(x)=φ 0 (x 0 )hλ,F(x)iγkF(x)k^2

That is

0  φ 0 (x 0 )(φ 0 (x)+hλ,F(x)i)γkF(x)k^2 =

= φ 0 (x 0 )L(x,λ)γkF(x)k^2.

Using the condition on the existence of the second derivative and the

stationarity conditionL^0 x(x 0 ,λ)=0 and that asx 0 is a feasible solution

L(x 0 ,λ)=φ 0 (x 0 )one must show that

0  (xx 0 )TL^00 xx(x 0 ,λ)(xx 0 )+o



kxx 0 k^2



γ

(^)
F^0 (x 0 )(xx 0 )+o(kxx 0 k)
(^2)
.