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PS3sol8.pdfProblem Set 3 Problem 1 Part
PS3sol8.pdfProblem Set 3 Problem 1 Part
PS3sol8.pdfProblem Set 3 Problem 1...
PS3sol8.pdfProblem Set 3 Problem 1 Part
Page 5
Pr
(
✓
=1

m
=
;
)=
Pr
(
m
=
;

✓
= 0)
Pr
(
✓
= 1)
Pr
(
m
=
;
)
=
1
/
10
1
/
11
2
/
11
=
1
20
As all other types do the same, the probability is the same for all types that are not 1:
Pr
(
✓
=
x

m
=
;
)=
1
/
20
for
x
=1
, ...,
10
.
Now we compute the expected value of
✓
conditional on receiving no message:
E
[
✓

m
=
;
]
=
0
Pr
(
✓
=0

m
=
;
)+1
Pr
(
✓
=1

m
=
;
) + 10
Pr
(
✓
= 10

m
=
;
)
=
1
1
20
+
2
20
+
3
20
+
...
+
10
20
=
55
20
=2
.
75
This cannot be an equilibrium because a ﬁrm with type
✓
=1
or
✓
=2
would prefer to send
no message, given that the expected value of their quality is higher (
2
.
75
vs.
1
or
2
).
We now do the same for
✓
⇤
=2
:
Pr
(
m
=
;
)=
2
11
+
9
110
=
29
110
Pr
(
✓
=0

m
=
;
)=
Pr
(
✓
=1

m
=
;
)=
10
29
Pr
(
✓
=2

m
=
;
)=
...P r
(
✓
= 10

m
=
;
)=
1
29
E
[
✓

m
=
;
]=1
10
29
+2
1
29
+
...
+ 10
1
29
=
64
29
=2
.
2
In this case, ﬁrms with
✓
=2
have incentives to send an empty message instead of disclosing
their type.
The following table shows the same calculations for all possible
✓
⇤
. There is an equilibrium
at
✓
⇤
=3
.
Firms do not disclosing their type (
✓
<
3)
do not have incentives to deviate
because with no message the expected value of their type is larger than their type. Firms
disclosing their type (
✓
≥
3
) do not want to send the empty message because the expected
value of quality given no message is lower than their type.
5
Page 6
✓
⇤
Pr
(
m
=
✓
)
Pr
(
✓
=
x

m
=
;
)
E
[
✓

m
=
;
]
✓
<
✓
⇤
✓
≥
✓
⇤
1
2
/
11
1
/
2
1
/
20
2
.
75
2
29
/
110
10
/
29
1
/
29
2
.
2
3
38
/
110
10
/
38
1
38
2
.
16
4
47
/
110
10
/
47
1
/
47
2
.
3
5
56
/
110
10
/
56
1
/
56
2
.
6
6
65
/
110
10
/
65
1
/
65
2
.
9
7
74
/
110
10
/
74
1
/
74
3
.
3
8
83
/
110
10
/
83
1
/
83
3
.
7
9
92
/
110
10
/
92
1
/
92
4
.
1
10
101
/
110
10
/
101
1
/
101
4
.
6
Problem 6
An attorney who just passed the bar is thinking of opening a sole proprietorship. He is either
skilled or not. A potential customer cannot observe the skill and thinks the two types are
equally likely. The customer can either hire the attorney or not and has the following payo
ﬀ
s:
•
u
C
(
hire, skilled
) = 10
;
u
C
(
hire, unskilled
)=

20
•
u
C
(
notHire, skilled
)=
u
C
(
notHire, unskilled
)=0
If a skilled attorney is hired, he proves himself, gets repeat business and receives a proﬁt of
50
. If an unskilled attorney is hired, he preforms poorly, doesn’t get hired again, but still
earns a proﬁt of
10
from the initial job. Both types of attorney get a proﬁt of zero if they
are not hired.
The attorney has to choose what type of sign to put on his new business. He can either put
up a cheap sign that costs 5 or a fancy sign that costs 15.
His overall payo
ﬀ
is his proﬁt
minus the cost of the sign.
Is there an equilibrium where skilled attorneys get hired?
Is there an equilibrium where
unskilled attorneys get hired?
Solution
There is a separating equilibrium in which only skilled attorneys get hired, and a pooling
equilibrium in which non of them gets hired. There is no equilibrium in which the unskilled
6
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