Economics 101, Problem Set 5
Solutions
Alan C. Marco
December 10, 2007
1.
Give all the Nash Equilibria for the following games. For each game, explain whether the NE (if any) is socially
optimal. Explain your answer.
(a)
Drop Dead Fred
Equilibrium: (Ouch, Sleep)
Fred
Phoebe
Cry
Sleep
Fly
1
;
1
6
;
&
1
7
;
0
Swim
2
;
0
Ouch
3
;
&
(b)
x;
i.e., &If
x < blah;
then the NE are
blank:
If
x
=
blab;
then.
..±]
If
x >
6 :
(Strange, Bottom)
If
x
±
6 :
(Up, Charm), (Strange, Bottom)
Amber
Up
6
;
1
6
;
6
3
;
4
Strange
x;
2
x;
0
4
;
3
Down
2
;
7
5
;
9
4
;
4
2.
Suppose two ²rms compete according to Cournot competition. They can produce in discrete units only. If
demand is given by
p
= 20
&
Q
and total costs for each ²rm are given by
c
(
q
) =
q
2
(so that
mc
= 2
q
with no
(a)
Compare the Cournot Nash equilibrium outcome to the ²rst best and to the monopoly (collusive) outcome.
[You may reproduce a partial payo/ matrix if you wish; i.e., you only need to show enough to calculate
the equilibria and the other relevant quantities. Hint: ²gure out the ²rst best ²rst so that you know the
maximum quanitity. Show at least one less than the monopoly output as well, in order to show that it
really is the monopoly outcome.]
0
1
2
4
5
6
0
0,0
0,18
0,32
0,48
0,50
0,48
1
18,0
17,17
16,30
14,44
13,45
12,42
2
32,0
30,16
28,28
24,40
22,40
20,36
1 3
42,0
39,15
36,26
30,36
27,35
24,30
4
48,0
44,14
40,24
32,32
28,30
24,24
5
50,0
45,13
40,22
30,28
25,25
20,18
6
48,0
42,12
36,20
²
The Cournot equilibrium is given by (4,4) for payo/s of (32,32).
²
The collusive outcome would be (3,3) for total payo/s of 66 (no other cell has a higher total payo/,
although it&s equivalent to 4,3 and 3,4).
²
The ±rst best is where price equals marginal cost for each ±rm, so the ±rst best is at (5,5): price is 10 and
mc is 10.
(a)
Explain how a cartel between the two ²rms would break down. Explain how your answer involves an
externality.
1
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Beginning at (3,3), each &rm has an incentive to cheat. Given that &rm 2 is playing ±3², then &rm
1³s best response is ±4.²Ditto for &rm 2.
&
By cheating on the agreement, each &rm can increase its pro&ts by 3.
&
But, this cheating leads to a negative externality: the other &rm³s pro&ts decrease. It³s an externality
because &rm 1³s private costs and bene&ts don³t take into account &rm 2³s loss.
(b)
Explain why &rms do not make zero pro&t at the &rst best.
This is because we³ve limited entry to two &rms. If we allowed for entry, &rms would enter. But, there
are no &xed costs, and
MC
is upward sloping. This means that
AC
is at its minimum at
q
= 0
:
So entry
will occur until we have an in&nite number of &rms each producing an in&nitesimal amount, so
and
price will be driven to zero, and quantity will be driven to 20. Not really realistic is it?
3.
Suppose that the demand for a product is given by
p
= 100
±
Q
d
and that the market supply is given by
p
=
Q
s
.
Unfortunately, production leads to pollution. Suppose that we can value the cost of the pollution in terms of
dollars. In particular, the marginal cost of pollution
(
e
)
is given by
e
=
Q;
so that the marginal cost of pollution
is increasing with production.
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 Spring '08
 Marco
 Economics, Microeconomics, Game Theory, Public Good, Market failure, Farmer, Externality

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