Economics 206
Spring 2007 (Prof. G. Loury)
Assignment 8: Signalling Games
(1)
Solution for the Grade In
f
ation Problem
:
You were asked to show that there is an “equilibrium without grade in
f
ation” in the
Grade In
f
ation Signalling Game if and only if:
[
θ
H
−
θ
L
]
≥
α
β
.
You were also asked to show that, when this condition fails, then there is a separating
equilibrium with grade in
f
ation, but only the professors of high ability students issue in
f
ated
evaluations.
An “equilibrium without grade in
f
ation” is de
F
nedasastrategyforeachtypeo
fpro

fessor,
r
=
r
∗
(
θ
)
, and market beliefs,
μ
=
μ
∗
(
r
)
, such that
∀
θ
∈
{
θ
L
,
θ
H
}
:
(i)
r
∗
(
θ
)=
θ
;
such that
∀
r
≥
0
:
(ii)
μ
∗
(
r
)
∈
[0
,
1]
,and
μ
∗
(
θ
L
)=0=1
−
μ
∗
(
θ
H
)
;
such that
∀
r
≥
0
:
(iii)
w
(
r
)=
μ
∗
(
r
)
θ
H
+(1
−
μ
∗
(
r
))
θ
L
;
andsuchthat
∀
r
≥
0
,
∀
θ
∈
{
θ
L
,
θ
H
}
:
(iv)
αθ
≥
α
w
(
r
)
−
β
(
r
−
θ
)
2
.
Condition (i) above says that no professor in
f
ates his grade. Condition (ii) says that
beliefs are consistent with Bayes’s Rule whenever possible. Condition (iii) says that, given
beliefs, wages equal expected productivity. And, condition (iv) says that for both types of
professor, given the market’s beliefs and the associated wages, the level of utility without
grade in
f
ation [
u
(
w
(
θ
)
,
θ
;
θ
)=
αθ
] is at least as great as the level of utility from reporting
any other grade
r
[
u
(
w
(
r
)
,r
;
θ
)=
α
w
(
r
)
−
β
(
r
−
θ
)
2
].
It follows immediately from the conditions above [taking
θ
=
θ
L
and
r
=
θ
H
in (iv)]
that in any nogradein
f
ation equilibrium:
(v)
αθ
L
≥
αθ
H
−
β
(
θ
H
−
θ
L
)
2
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 Spring '07
 G.LOURY
 Economics, Microeconomics, Game Theory, Political correctness, Bayesian probability

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