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PS3sol8.pdf
PS3sol8.pdf
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PS3sol8.pdf-Problem Set 3 Problem 1 Part
PS3sol8.pdf-Problem Set 3 Problem 1 Part
PS3sol8.pdf-Problem Set 3 Problem 1...
PS3sol8.pdf-Problem 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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