b
) Does
u
attain a maximum on the set
B
=
{
(
x,y
) :
y
≥
0,
px
+
y
≤
3
}
? Why?
Answer:
This is a trickier problem. As was the case with the cost minimization problem done
in class, we must simplify the problem before solving it. First, note that
u
is increasing in
both
x
and
y
. It follows that any maximum must occur along the constraint
px
+
y
= 3. So
x
= (3

y
)
/p
. Substituting into
u
, we reduce our problem to maximizing (3

y
)
/p
+
√
y
over
the set
y
≥
0. Note that the maximum must be at least 3
/p
, which is the value taken at
y
= 0.
It is easy to see that lim
y
→∞
(3

y
)
/p
+
√
y
=
∞
, so there is some
b
with (3

y
)
/p
+
√
y <
3
/p
for
y > b
. We can now remove any
y > b
from the set we are maximizing over, and focus on
whether (3

y
)
/p
+
√
y
can be maximized over [0
,b
]. But now we are maximizing a continuous
function over a compact set, and Weierstrass’s Theorem tells us that there is a maximum. This
is also a maximum for the original problem.
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 Spring '08
 STAFF
 Economics, Topology, Metric space, λ, Closed set, mathematical economics midterm

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