Econ 6202, Fall 2006
Dmitry Shapiro
Final Exam
December 12, 2006
•
ATTENTION! THIS IS 150 MINUTE EXAM!
•
There are FIVE questions in the final. You need to do ALL of them.
•
The exam is 150 minutes long and has 150 points.
Each question has 30 points assigned.
Thus it would make sense for you to start with easier questions.
•
The exam is
closedbook.
•
Please turn off your cellphones and other soundmaking devices.
Good luck!
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150 minute
exam today
(December 12).
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75 minutes today
and 75 minutes on Thursday
. You can pick any that suits you best.
This is 150 minute exam!
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1
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1.
[Producer Theory and Monopoly  30 points]
Consider the following economy. There
are two consumers, 1 and 2; who derive utility from a certain good
y
and from money
m
.
The consumers utility functions are
u
1
(
x, m
) = 36
y
+
m,
u
2
(
x, m
) = 144
√
y
+
m.
A monopolist produces the good
y
by using two inputs:
x
1
and
x
2
. The production function
is
y
=
√
x
1
+
√
x
2
.
Let
w
i
denote the price of input
i
= 1
,
2.
(a) Compute the monopolist’s cost function.
(b) Suppose
w
1
= 6,
w
2
= 3.
Suppose also that the monopolist observes the consumers
utility and makes takeitorleaveit offers (
r
i
, y
i
),
i
= 1
,
2 (
r
i
denotes the payment that
consumer
i
has to make to receive
y
i
units of the good).
Compute the optimal offers
(i.e. the offers that maximize the monopolist’s profits).
2.
[General Equilibrium  30 points]
In a twoconsumer, twogood exchange economy the
consumers have the following preferences:
u
1
(
x
1
, x
2
) = min
{
x
1
,
2
x
2
}
,
u
2
(
x
1
, x
2
) =
x
1
/
2
1
x
1
/
2
2
.
(1)
The endowments of consumer 1 and 2 are
e
1
= (5
,
9) and
e
2
= (12
,
1), respectively.
Suppose that the government of this economy can redistribute wealth between the two con
sumers by choosing
monetary
transfers. Let
T
denote the transfer from consumer 1 to con
sumer 2. If
T >
0 then consumer 1 gives
T
dollars to consumer 2. If
T <
0 then consumer 2
gives
T
dollars to consumer 1.
Given a transfer
T
, let
E
(
T
) denote the following economy. Consumer 1 owns the bundle
e
1
and

T
dollars. Consumer 2 owns the bundle
e
2
and
T
dollars. Consumers’ preferences are
described by (1).
(a) Consider the allocation
x
= (
x
1
, x
2
) = ((8
,
4)
,
(9
,
6)).
Can you find a transfer
T
such
that
x
is the Walrasian equilibrium allocation of the economy
E
(
T
)? If your answer is
affirmative, compute the value of
T
and of the Walrasian equilibrium prices (fix
p
1
= 1).
If your answer is negative, explain why such a transfer does not exist.
(b) Redo question 2a with the allocation ¯
x
= ((7
,
3)
,
(10
,
7)).
3.
[General Equilibrium II  30 points]
An exchange economy has two consumers,
A
and
B
, and two gods
a
and
b
. Consumers’ utility is given by
U
A
(
a, b
) =
√
a,
U
B
(
a, b
) =
√
b.
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 Fall '09
 SHAPIRO
 Economics, Game Theory, Utility, Consumer, ﬁrst consumer

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