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Winter 2011
Solutions for Problem Set VI
1.a.
Solve the constraint to get
y
mx
and substitute for
y
in the objective function to see that we
want to choose
x
to maximize or minimize
2
()
zxmx m
xx
.
The first order condition is
/2
0
dz dx
m
x
which implies that
xm
.
Substitute this into the constraint to get
ym
.
Since
22
0
dz d
x
,
and
/ 2
gives a maximum.
1.b
The Lagrangean for this problem is
(
)
x
x
y
L
.
The first order conditions are
0
xy
L/
,
0
yx
, and
0
mxy
.
The first two imply that
x
y
, and we can use this and the constraint to see that
and
/ 2
.
The bordered
Hessian is
21
31
01
1
11
1
1
1 0
1 1
(1
)
1
)
11 2 0
10
0 1
0
xx
xy
x
yx
yy
y
LLL
where
the second equality follows by evaluating the determinant down the first column.
It follows that
and
/ 2
gives a maximum.
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 Spring '08
 Ellis,G

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