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Unformatted text preview: from the 3rd
∗∗
condition, will also satisfy our required constraint,
,
0.
But we still have a possible unresolved issue at hand – while the critical points
∗∗∗
, , may maximize or minimize (in a local sense) the function
, , , how do
,
?
we know we can say the same regarding the original function, Let’s think about the way to approach this question. First, we want to consider how
, , changes with changes in
, , , i.e., We know that the solution to 13 above, ∗ , ∗ , ∗ is a critical point of
,
0 at ∗ , ∗ , ∗ . We want to take this information and to show that
0 for all permissible changes in
,
ensure , , so
0 will
. By permissible changes, we mean precisely those that conform to our constraint,
,
0.
Notice that at ∗ , ∗ , ∗ , the total derivative, can be rewritten as:
∗ ∗ ,
∗ ∗ ∗ , ∗ , ∗ ∗
∗ , ∗ This expression is obtained by solving out for the partials
above. ∗ , 0
, , and , using 1 – 3 We now want to consider permissible movements in
and
and
that conform to the constraint,
,
are changes in
, , , ; as we noted these
0. We have:
, ∗∗
which, when evaluated at ,
,
, equals 0. Note that is the last term in
the parenthesis in the above expression for
. (What we are doing is adjusting
and
along a path that ensures
,
0 for all points along the path. That is,
this is the only kind of changes in the two variables we consider, since our problem
places the constraint that
,
0 must hold).
∗ Also note that at the critical point,
that
∗ , , ∗ 0 (see 3 above). Together, these imply ∗ But this gives us our required result. It says ∗ , ∗ 0
0, meaning that ∗ , ∗ will be a critical point for the original function given that the variables
must satisfy
,
0 and any movement away from ∗ , ∗ must be along a
path that conforms to this constraint. An intuitive look at constrained optimization (Lagrange Revisited)
Suppose we have a level set
are to remain on that level set,
, , ,
:
,
. If we change , and we
must be such that is unchanged, and equal to : , 0. This is true along any level set of the function. Therefore,
 Of course, this assume...
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This document was uploaded on 02/07/2014.
 Winter '14
 Economics

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