Economics 206
Spring 2007 (Prof. G. Loury)
Assignment 8: Signalling Games
(1) Consider the following
Grade InFation Problem
: A professor must issue an evaluation
of his student to “the market.” The professor perfectly observes his student’s productivity,
but the market can only observe the professor’s evaluation. Let
θ
∈ {
θ
L
,θ
H
} ⊂
[0
,
∞
) be a
student’s two possible levels of productivity – either “high” or “low,” and let
r
∈
[0
,
∞
) be
a professor’s evaluation of student ability, which can be any nonnegative number.
Suppose that the market rewards a student according to how productive that student is
believed to be, and that market beliefs about a student’s productivity are derived from the
professor’s evaluation. SpeciFcally, let the market reward be denoted by
w
(
r
), and assume
that:
w
(
r
) =
E
[
θ

r
]
.
In addition, assume that professors care both about the rewards received about their stu
dents, and about the accuracy of their evaluation. In particular, the professor derives utility
from seeing a student get a higher market reward but experiences disutility from issuing
an evaluation of a student which di±ers from the student’s true productivity. Denoting a
professor’s utility by
u
(
w,r
;
θ
), assume the particular funcitonal form:
u
(
w,r
;
θ
) =
αw

β
(
r

θ
)
2
,
where
α >
0 and
β >
0 are Fxed preference parameters.
²inally, suppose that Pr
{
θ
=
θ
H
}
=
λ
∈
(0
,
1) is the prior probability that a student’s
productivity is high. We can think of this situation as a signalling game where a professor can
be of either of two types (i.e., either with a high productivity or a low productivity student)
and must issue an evalution. Let the professor follow a strategy in doing so,
r
=
r
*
(
θ
), which
indicates the evaluation that is issued for a student of each ability level. Moreover, let the
market forms beliefs
μ
=
μ
*
(
r
), taking these strategies into account and using Bayes’s Rule
where possible, but otherwise in an unrestricted way. ²inally, let rewards to students be
equal to the student’s expected productivity given these beliefs. Thus:
w
*
(
r
) =
μ
*
(
r
)
θ
H
+ [1

μ
*
(
r
)]
θ
L
.
Now, consider the possible equilibria in this signalling model. An equilibrium is characterized