[BUG] lagrange multipliers and econ

[Matthias Kawski]

On Tue, 22 Feb 2005, Smith, Alexander J. wrote:

> It would seem that Lagrange multipliers do make sense, even if
> the gradient does not. To find a critical point to z=f(x,y)
> subject to the constraint g(x,y)=c more or less boils down to
> the observation that if the level curve of f cuts the curve
> g(x,y)=c transversally, ....

The way I understand this is that for constrained minimization
problems (in the spirit of Lagrange multipliers) the issue is
linear (in)dependence, but angles do not matter. Either some
(co)vector is in the linear span (like: angle=zero) or it is
not (like: angle=nonzero). [[This becomes clearer in dim three
and higher, e.g. with two constraints g1=0 and g2=0 and trying
to minimize f.]]
Therefore it should not matter whether one formulates this in
terms of differentials or gradients, i.e. I'd expect ALL the
pictures with "parallel"/"tangential"/"transversal" which you
were describing are intrinsically sound.


Matthias
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Matthias Kawski                http://math.asu.edu/~kawski
Dept. of Mathematics and Statistics         kawski at asu.edu
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