Econ 101A — Midterm 2
Th 2 November 2006.
You have approximately 1 hour and 20 minutes to answer the questions in the midterm. I will collect
the exams at 11.00 sharp. Show your work, and good luck!
Problem 1. Pro
f
t Maximization with Uncertainty
(65points)AllisonownsacompanyupinNapa
that produces wine. Her income is given by the pro
f
ts of the company that she runs. Allison is riskaverse,
with utility function
u
(
c
)
satisfying
u
0
(
c
)
>
0
for all
c,
and
u
00
(
c
)
<
0
for all
c.
Allison maximizes her utility
from the pro
f
ts of the company. Wine
y
is a laborintensive operation. It is produced using labor with the
production function
y
=
l
α
,
with
0
<α<
1
.
Workers in the vine are paid the wage
w.
Allison maximizes
utility from her only income source, the pro
f
ts from wine sales:
max
l
u
[
pl
α
−
wl
]
.
(1)
1. De
f
ne the concept of returns to scale, that is, what we mean by constant/increasing/decreasing returns
to scale. Under what conditions for
α
the production of wine has decreasing returns to scale? Show
the steps of your reasoning. (Here assume only
α>
0
,
from the next point on assume
0
<α<
1
)(4
points)
2. Consider the maximization of utility of Allison, and obtain
f
rstorder conditions.(3 points)
3. Are the second order conditions satis
f
ed? (3 points)
4. Solve for the utilitymaximizing
l
∗
.
How does the optimal employment of workers in the vineyard vary
as the wage
w
of workers increases? And when the price
p
increases? Discuss the economic intuition.
(3 points)
5. Using the envelope theorem, compute the e
f
ect on the utility of Allison at the optimum
u
[
p
(
l
∗
)
α
−
wl
∗
]
of an increase in price
p.
Does the
f
nding make sense? Discuss. (4 points)
6. In a nearby terrain, Wei runs his operation with a standard pro
f
tmaximization. He has the same
production function and costs as Allison, and hence maximizes
max
l
pl
α
−
wl.
(2)
So
lveforthepro
f
tmaximizing
l
∗
for Wei. (3 points)
7. Allison and Wei are neighbors to a chatty economist trained in Stanford. (a concession to Prasad) The
economist, after drinking a little too much of their wine, is ontherecord on the local TV as saying
“it is a pity that Allison is so riskaverse. Unlike Allison, Wei just maximizes pro
f
ts, which is much
better for the company”. Discuss this assertion in light of what you found so far. Provide economic
and mathematical intuition. (6 points)
8. The world of winegrowing is a treacherous one, with much uncertainty. In particular, the price of wine
F
uctuates from yeartoyear, and the decisions on how much to produce and how many workers to hire
have to be made before the level of price is known. The price
p
of a bottle is
10
with probability
q
and
5
with probability
1
−
q.
Let’s consider
f
rst the problem of Wei, (2). Wei now maximizes
expected
pro
f
ts. Set up the maximization of expected pro
f
ts by Wei, explaining why you are setting up the
problem the way you are. (6 points)
9. Derive Wei’s
f
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 Spring '08
 Staff
 Utility, Wei, Allison l∗

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