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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 purestrategy 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
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View Full DocumentWe then compute the Best Response correspondence for player 2. Player 2 prefers Left to Right if
u
∗
u
2
(
U, L
)+(1
−
u
)
u
2
(
D,L
)
≥
u
∗
u
2
(
U, R
−
u
)
u
2
(
D,R
)
or
2
u
+(1
−
u
)
≥
u
+3(1
−
u
)
or
u
≥
2
/
3
.
Therefore, the Best Response correspondence for player 2 is
BR
∗
2
(
u
)=
⎧
⎨
⎩
l
=1
if
u>
2
/
3;
any
l
∈
[0
,
1]
if
u
=2
/
3;
l
=0
if
u<
2
/
3
.
Plotting the two Best Response correspondences, we see that the three points that are on the Best Re
sponse correspondences of both players are
(
σ
∗
1
,σ
∗
2
)=(
u
,l
=1)
,
(
u
=0)
,
and
(
u
/
3
/
3)
.
The
f
rst two are the purestrategy equilibria we had identi
f
ed before, the other one is the additional
equilibrium in mixed strategies.
2. We proceed casebycase:
(a)
C
0
(
q
C
(
q
)
/q
.
The marginal cost function is always (weakly) above the average cost
function). Supply function:
S
(
p
⎧
⎨
⎩
q
∗
→
+
∞
if
p>
2
any
q
∈
[0
,
∞
)
if
p
q
∗
if
p<
2
(b)
C
0
(
q
)=4
q
−
1
,C
(
q
)
/q
q
−
1+2
/q.
Marginal cost is higher than average cost whenever
4
q
−
1
≥
2
q
−
/q,
or
2
q
2
−
2
≥
0
,
or
q
2
≥
1
,
or
q
≥
1
.
(we do not care about solutions with
negative
q
)P
lugin
q
in the marginal cost curve to
f
nd the lowest price level such that the
marginal cost function lies above the average cost function:
p
=4
∗
(1)
−
1
,
or
p
=3
.
We invert
the marginal cost function
C
0
(
q
q
−
1=
p
to get
q
=
p/
4+1
/
4
.
The supply function therefore
is
S
(
p
½
q
∗
=
p/
/
4
if
p
≥
3
q
∗
if
3
3.
C
0
(
q
)=3
q
2
+10
(
q
)
/q
=
q
2
.
Marginal cost is higher than average cost whenever
3
q
2
≥
q
2
,
or
2
q
2
≥
0
,
which is always true. We invert the marginal cost function
C
0
(
q
q
2
+10=
p
to get
q
=
q
(
p
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 Spring '08
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