Econ 101A — Solutions to Final Exam
Th 15 December.
Please solve Problem 1, 2 and 3 in the
f
rst blue book and Problems 4 and 5 in the second Blue Book.
Good luck!
Problem 1. Shorter problems.
(50 points) Solve the following shorter problems.
1. Compute the pure-strategy and mixed strategy equilibria of the following coordination game. Call
u
the probability that player 1 plays Up,
1
−
u
the probability that player 1 plays Down,
l
the probability
that Player 2 plays Left, and
1
−
l
the probability that Player 2 plays Right. (20 points)
1
\
2
Left
Right
Up
3
,
21
,
1
Down
1
,
12
,
3
2. For each of these cost functions, plot the marginal cost function and the supply function,
and
write
out the supply function
S
(
p
)
,
with quantity as a function of price
p
(30 points):
(a)
C
(
q
)=2
q
(8 points)
(b)
C
(
q
q
2
−
q
+2
(12 points)
(c)
C
(
q
)=
q
3
+10
q
(10 points)
Solution to Problem 1.
1. The pure strategy Nash equilibria can be found in the matrix once we underline the best responses for
each player:
1
\
2
Left
Right
Up
3
,
2
1
,
1
Down
1
,
,
3
The equilibria therefore are
(
s
∗
1
,s
∗
2
)=(
U, L
)
and
(
s
∗
1
∗
2
D,R
)
.T
o
f
nd the mixed strategy
equilibria, we compute for each player the expected utility as a function of what the other player does.
We start with player 1. Player 1 prefers Up to Down if
lu
1
(
U, L
)+(1
−
l
)
u
1
(
U, R
)
≥
lu
1
(
D,L
−
l
)
u
1
(
)
or
3
l
+(1
−
l
)
≥
l
+2(1
−
l
)
or
l
≥
1
/
3
.
Therefore, the Best Response correspondence for player 1 is
BR
∗
1
(
l
⎧
⎨
⎩
u
=1
if
l>
1
/
3;
any
u
∈
[0
,
1]
if
l
/
3;
u
=0
if
l<
1
/
3
.
1