Econ 101A — Midterm 1
Th 28 February 2008.
You have approximately 1 hour and 20 minutes to answer the questions in the midterm. Vikram will
collect the exams at 11.00 sharp. Show your work, and good luck!
Problem 1. ThreeGood CobbDouglas.
(50 points) Seung likes three goods:
x
1
,
x
2
,and
x
3
.
He is
aware that in Econ 101A we only use two goods, but he is too attached to all of them to let go of one. He
maximizes the utility function
u
(
x
1
,x
2
3
)=
x
α
1
1
x
α
2
2
x
α
3
3
,
with
0
<α
i
<
1
for
i
=1
,
2
,
3
. The consumption good
x
i
has price
p
i
(for
i
,
2
,
3
) and the individual has
total income
M
.
1. Compute the marginal utility of consumption with respect to good
x
1
,
∂u
(
x
1
2
3
)
/∂x
1
.(2po
in
t
s
)
2. What is the limit of the marginal utility for
x
1
→
0
and for
x
1
→∞
?
Interpret the economic intuition
behind this feature of this utility function. (5 points)
3. Write the budget constraint. (3 points)
4. Write the maximization problem of Seung. Seung wants to achieve the highest utility subject to the
budget constraint. Write down the boundary constraints for
x
1
2
3
, and neglect them for now. (3
points)
5. Assuming that the budget constraint holds with equality, write down the Lagrangean and derive the
f
rst order conditions with respect to
x
1
2
3
,
and
λ.
(5 points)
6. Solve for
x
∗
1
as a function of the prices
p
1
,p
2
3
,
the income
M,
and the parameters
α
1
,α
2
,
and
α
3
.
[Hint: combine the
f
rst and second
f
rstorder condition, then combine the
f
rst and third
f
rstorder
condition, and
f
nally plug in budget constraint] Similarly solve for
x
∗
2
and
x
∗
3
.(6po
t
s
)
7. Is this true or false? Show: “CobbDouglas preferences have the feature that the share of money spent
on each good does not depend on the income, or on prices” (6 points)
8. Are the boundary conditions for
x
1
2
,
and
x
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 Spring '08
 Staff
 Economics, Utility, SeUng

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