Econ 51: Final Exam
June 7, 2004
Instructions:
•
This is a closed book, closednotes exam. You are allowed to use a handheld calculator
(i.e. no laptop).
•
You have 180 minutes.
•
Please answer all 5 questions. Please use a separate blue book for each question.
•
Always remember: think before you do the math.
•
Most questions start easy and get more di
ﬃ
cult, so make sure you organize your time
intelligently (5.d may be particularly hard).
•
Good luck!
1 Walrasian Equilibrium and Pareto E
ﬃ
ciency (23 points)
Consider an economy with two goods, apples (
a
) and bananas (
b
), two agents (Rob and
Tom). Rob’s utility is given by:
u
R
(
a
R
, b
R
) =
a
R
+ 2
b
R
Tom’s utility is given by:
u
T
(
a
T
, b
T
) = 2
a
T
+
b
T
Aggregate endowments are
e
a
= 100 and
e
b
= 40.
(a)
(5 points) Describe the set of all Pareto E
ﬃ
cient allocations in this economy.
Suppose now that this economy also has two
fi
rms, 1 and 2. Firm 1 can transform one
apple to one banana, i.e.
Y
1
=
{
(
y
a
, y
b
) :
y
a
≤
0
, y
b
=
−
y
a
}
.
Firm 2 can transform one
banana to one apple, i.e.
Y
2
=
{
(
y
b
, y
a
) :
y
b
≤
0
, y
a
=
−
y
b
}
.
(b)
(7 points) Describe the set of all Pareto E
ﬃ
cient allocations in this new economy
with
production
.
1
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(c)
(7 points) Suppose that Rob is endowed with all apples and no bananas, and Tom is
endowed with all bananas and no apples, i.e.
e
R
= (100
,
0) and
e
T
= (0
,
40). Compute
a Walrasian equilibrium: prices, consumption by Rob and Tom, and production plans
for the
fi
rms.
(d)
(4 points) Two new
fi
rms enter the market (
fi
rms 1 and 2 are still around). Firm 3 can
transform two apples to one banana, i.e.
Y
3
=
{
(
y
a
, y
b
) :
y
a
≤
0
,
2
y
b
=
−
y
a
}
. Firm
4 can transform two bananas to one apple, i.e.
Y
4
=
{
(
y
b
, y
a
) :
y
b
≤
0
,
2
y
a
=
−
y
b
}
.
Repeat parts (b) and (c).
2 Externalities (22 points)
Consider an economy with two agents (Rob and Tom) and two goods, cigarettes (
c
) and
money (
m
). A
fi
rm can transform one unit of money to one cigarette, i.e.
Y
=
{
(
y
m
, y
c
) :
y
m
≤
0
, y
c
=
−
y
m
}
. Rob and Tom both enjoy smoking, but su
ff
er when the other is smoking.
In particular, Rob’s utility is given by
u
R
(
c
R
, c
T
, m
R
) = 3 log(
c
R
)
−
log(
c
T
) +
m
R
and Tom’s utility is given by
u
T
(
c
T
, c
R
, m
T
) = 3 log(
c
T
)
−
log(
c
R
) +
m
T
(as always, logs are natural logarithm, i.e. base
e
). Rob and Tom are each endowed with 20
units of money and no cigarettes.
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 Winter '07
 Tendall,M
 Equilibrium, Game Theory, Tom, rob

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