ECON 400
Hartman
Winter 2011
Problem Set VI
(for Monday, February 14)
1.
Suppose the problem is to find the
x
and
y
that either maximize or minimize
z
xy
subject to the constraint
0
m
x
y
where
m
is a positive parameter.
a.
Solve this problem by the method of direct substitution.
(The
x
and
y
you find
will depend on
m
.)
Use the second order condition to determine whether the
x
and
y
you have found give a maximum or a minimum.
b.
Solve this problem using a Lagrange multiplier.
Use the second order
condition to determine whether the
x
and
y
you have found give a maximum
or a minimum.
c.
Let
*
(
)
x
m
and
*
(
)
y
m
denote the solution for
x
and
y
from part b, and let
*
(
)
m
denote the Lagrange multiplier from part b.
Substitute
*
(
)
x
x
m
and
*
(
)
y
y
m
into the expression to be maximized or minimized to get
*
*
*
(
)
(
)
(
)
z
m
x
m y
m
.
Show that
*
*
(
)
dz
m
dm
.
2.
Suppose the problem is to find the
x
and
y
that either maximize or minimize
z
x
y
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 Spring '08
 Ellis,G
 Optimization, order condition

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