Econ 407: Lecture on the application of the lagrangian multiplier to minimize cost function for a firm producing two goods, x and y. We use the first and second order conditions to demonstrate the application of the method. Bascially, you have to take thee partial derivatives, set them equal to zero, and solve for the three unknowns. After that, use second order conditions to check for the existence of a relative minimum.

> C:=8*x^2-x*y+12*y^2;

> con:=42-x-y;

> L:=C+lambda*(con);

> Lx:=diff(L,x);

> Ly:=diff(L,y);

> Llambda:=diff(L,lambda);

> solve({Lx,Ly,Llambda},{x,y,lambda});

Let us examine the second-order conditions.

> Lxx:=diff(Lx,x);

> Lyy:=diff(Ly,y);

> Lxy:=diff(Lx,y);

> Pdct:=Lxx*Lyy;

> Sqcross:=(Lxy)^2;

Since the products of the second derivatives is greater than the squared cross derivative and since Lxx>0, and Lyy>0, we have the conditions for a relative minimum.

> eval(C,{x=25,y=17});

> plot3d(C,x=0..50,y=0..40);

Second hourly exam Monday, Oct. 25, 1999. Don't Miss IT! There will be no make-ups.