Economics 101, Problem Set 5
Solutions
Alan C. Marco
December 7, 2006
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
Laugh
Cry
Sleep
Fly
1
,
1
6
,

1
7
,
0
Swim
2
,
0
5
,
0
8
,

1
Ouch
3
,

1
4
,
1
9
,
1
(b)
Quarks [Express your answer for di±erent values of
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
Buffy
Top
Charm
Bottom
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
²xed costs), create a payo± matrix for the game, and determine the Nash equilibrium(a).
(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.]
2
0
1
2
3
4
5
6
0
0,0
0,18
0,32
0,42
0,48
0,50
0,48
1
18,0
17,17
16,30
15,39
14,44
13,45
12,42
2
32,0
30,16
28,28
26,36
24,40
22,40
20,36
1 3
42,0
39,15
36,26
33,33
30,36
27,35
24,30
4
48,0
44,14
40,24
36,30
32,32
28,30
24,24
5
50,0
45,13
40,22
35,27
30,28
25,25
20,18
6
48,0
42,12
36,20
30,24
24,24
18,20
12,12
•
The Cournot equilibrium is given by (4,4) for payo±s of (32,32).
•
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).
•
The ²rst best is where price equals marginal cost for each ²rm, so the ²rst 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