Unformatted text preview: The firm’s costs
(or cost function) is given by the function
,
5
80
32 . The firm
30. Of course, we
wants to minimize its costs subject to a production constraint,
also have the non‐negativity constraints,
0 and
0.
We can translate this into one of a maximization problem by looking at
,
. The firm
then chooses
and
to maximize 5
80
32 , subject to the constraints
above.
The Lagrangian for this problem is:
5
80
32
30 .
Our first‐order conditions are:
: 10
80
0 10
80
0.
1.
2. : 32 0 32 0. 3. :
30
0 30
0.
When approaching a problem like this, first check out corner solutions for the choice
variables ( and ). Suppose
0. Then
80, which violates our K‐T condition that
0, we conclude
32. So we
0. So this can’t apply. Likewise, if we assume
can’t have corner solutions.
Need the constraint bind? Suppose not. Then
0 (the constraint doesn’t affect the
30 0. But from the second conditions
objective function at the margin) and so
of 1) and 2) (with
0) we see
8 and
32, which means the constraint would
bind. Hence,
0. 2. The Implicit Function Theorem
We have on a few occasions made use of a very useful theorem in economics, the implicit
function theorem. It is time to point it out explicitly.
Suppose we had a nice, function which is continuous and has continuous first partial
derivatives, where
,
0 at some point ∗ , ∗ in its domain. We’ll keep things
somewhat general here as to what
,
, , and the condition
,
0 may
represent.
∗∗
If we assume
,
0, the implicit function states that there exists a neighborhood of
∗∗
,
such that any value of corresponds to a unique value of
and hence, we can
write as a continuous function of
,
. Moreover, this function is
differentiable. More generally, consider the system...
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 Winter '14
 Economics, Derivative, producer, objective function, Constraint, Gradient, quasiconcavity

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