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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
zx
y
subject to the constraint
0
mxy
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 xm
and
*
y ym
into the expression to be maximized or minimized to get
**
*
() ()
zm xmym
.
Show that
*
*
dz
m
dm
.
2.
Suppose the problem is to find the
x
and
y
that either maximize or minimize
zxy
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This note was uploaded on 04/04/2011 for the course ECON 400 taught by Professor Ellis,g during the Spring '08 term at University of Washington.
 Spring '08
 Ellis,G

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