Economics 206
Spring 2007 (Prof. G. Loury)
Assignment 8: Signalling Games
(1) Consider the following
Grade Inflation 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. Specifically, 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 differs 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 fixed preference parameters.
Finally, 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.
Finally, 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
.
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 Spring '07
 G.LOURY
 Economics, Microeconomics, Government, Public Policy, unbiased expert

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