econ/ams 11b
winter 2010
Midterm 2 Review
1.
FInd the critical point(s) of the functions below subject to the given constraint.
a.
h
(
u, v
) = 5
u
+ 8
v
;
2
u
2
+ 3
v
2
= 30.
b.
k
(
x, y, z
) =
xy
2
z
3
;
5
x
+ 8
y
+ 12
z
= 300.
2.
A household’s utility function is given by
U
(
q
A
, q
B
, q
C
) = 5 ln
q
A
+12 ln
q
B
+18 ln
q
C
, where
q
A
,
q
B
and
q
C
are the quantities of goods of type
A
,
B
and
C
that the household consumes
per year. The average prices per unit of these goods are
p
A
= 10,
p
B
= 15 and
p
C
= 25,
respectively, and the household’s disposable income is
Y
d
= $60
,
000.
a.
Assuming that the household spends all of its disposable income on these three goods,
how many units of each type should the household consume to
maximize its utility
?
b.
By how much will the household need to increase its disposable income to increase its
maximum utility by 5 units? Justify your answer.
c.
What proportion of its income does the household spend on each type of good when
utility is maximized? Do these proportions depend on the income level? Justify your
answer.
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 Spring '10
 yk
 Economics, Utility, Disposable income, $1000, $60,000, $2500, $5000.00

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