LECTURE NOTES ON SPENCE’S JOB
MARKET SIGNALLING
1
SIMPLIFIED MODEL
2
SIMPLIFIED MODEL
The Set-Up
Two “types” of workers: HIGH ability (θ = 2),
and LOW ability (θ = 1), where θ measures
ability.
SIMPLIFIED MODEL
The Set-Up
Two “types” of workers: HIGH ability (θ = 2),
and LOW ability (θ = 1), where θ measures
ability.
Employers don’t know the type of any one
worker but have commonly known prior beliefs:
Pr(θ = 1) = 31 , and Pr(θ = 2) = 23 .
SIMPLIFIED MODEL
The Set-Up
Two “types” of workers: HIGH ability (θ = 2),
and LOW ability (θ = 1), where θ measures
ability.
Employers don’t know the type of any one
worker but have commonly known prior beliefs:
Pr(θ = 1) = 31 , and Pr(θ = 2) = 23 .
Productivity of worker is 2θ
SIMPLIFIED MODEL
The Set-Up
Two “types” of workers: HIGH ability (θ = 2),
and LOW ability (θ = 1), where θ measures
ability.
Employers don’t know the type of any one
worker but have commonly known prior beliefs:
Pr(θ = 1) = 31 , and Pr(θ = 2) = 23 .
Productivity of worker is 2θ
Cost of education e is C (e) = θe .
SIMPLIFIED MODEL
The Set-Up
Two “types” of workers: HIGH ability (θ = 2),
and LOW ability (θ = 1), where θ measures
ability.
Employers don’t know the type of any one
worker but have commonly known prior beliefs:
Pr(θ = 1) = 31 , and Pr(θ = 2) = 23 .
Productivity of worker is 2θ
Cost of education e is C (e) = θe .
Signalling game: First, the worker chooses the
level of eduction, e. The employer, upon observing e, chooses wage.
PERFECT BAYESIAN EQUILIBRIA.
3
PERFECT BAYESIAN EQUILIBRIA.
Simplify Analysis: Assume wage equals to expected productivity.
PERFECT BAYESIAN EQUILIBRIA.
Simplify Analysis: Assume wage equals to expected productivity.
Separating PBE
Can a Separating Perfect Bayesian Equilibrium
exist?
Suppose it does. Then it must be the case
that:
In a separating PBE the two types of workers
choose different education levels: Let e H and
e L denote the levels chosen by high and low
ability types, respectively, where e H 6= e L .
PERFECT BAYESIAN EQUILIBRIA.
Simplify Analysis: Assume wage equals to expected productivity.
Separating PBE
Can a Separating Perfect Bayesian Equilibrium
exist?
Suppose it does. Then it must be the case
that:
In a separating PBE the two types of workers
choose different education levels: Let e H and
e L denote the levels chosen by high and low
ability types, respectively, where e H 6= e L .
Furthermore, the posterior beliefs of employers
in such a separating PBE will be as follows:
Apply Bayes rules (when can, following events
that have non-zero probability of occurring):
Pr(θ = 2 | e = e H ) = 1 and Pr(θ = 1 | e = e L ) = 1
4
Apply Bayes rules (when can, following events
that have non-zero probability of occurring):
Pr(θ = 2 | e = e H ) = 1 and Pr(θ = 1 | e = e L ) = 1
Cannot apply Bayes rule following zero probability events — i.e., in the separating PBE
when education level e is observed different
from e H and e L .
Apply Bayes rules (when can, following events
that have non-zero probability of occurring):
Pr(θ = 2 | e = e H ) = 1 and Pr(θ = 1 | e = e L ) = 1
Cannot apply Bayes rule following zero probability events — i.e., in the separating PBE
when education level e is observed different
from e H and e L .
Indeed, thus, for any e such that e 6= e H and
e 6= e L : Pr(θ = 1 | e) can be any number between zero and one. The PBE concept does
not restrict out-of-equilibrium beliefs.
Apply Bayes rules (when can, following events
that have non-zero probability of occurring):
Pr(θ = 2 | e = e H ) = 1 and Pr(θ = 1 | e = e L ) = 1
Cannot apply Bayes rule following zero probability events — i.e., in the separating PBE
when education level e is observed different
from e H and e L .
Indeed, thus, for any e such that e 6= e H and
e 6= e L : Pr(θ = 1 | e) can be any number between zero and one. The PBE concept does
not restrict out-of-equilibrium beliefs.
Suppose, then, (to most easily see whether a
separating PBE exists), assume: Pr(θ = 1 | e) =
1 for any e such as e 6= e H and e 6= e L .
That is: we assume that when employers observe education e 6= e H , they believe worker is
Low type for sure.
Given the above, the wages in this PBE must
be as follows (since assumed above wages equal
expected productivity):
w(e = e H ) = 2(2) = 4 and for any e 6= e H , w(e) =
2(1) = 2.
5
Given the above, the wages in this PBE must
be as follows (since assumed above wages equal
expected productivity):
w(e = e H ) = 2(2) = 4 and for any e 6= e H , w(e) =
2(1) = 2.
INCENTIVE COMPATIBILITY CONDITIONS
HIGH Type’s IC conditions:
For any e 6= e H ,
e
e
4− H ≥ 2− .
2
2
Given the above, the wages in this PBE must
be as follows (since assumed above wages equal
expected productivity):
w(e = e H ) = 2(2) = 4 and for any e 6= e H , w(e) =
2(1) = 2.
INCENTIVE COMPATIBILITY CONDITIONS
HIGH Type’s IC conditions:
For any e 6= e H ,
e
e
4− H ≥ 2− .
2
2
This implies the High type IC condition becomes:
e
4 − H ≥ 2.
2
Consequently, for the proposed separating
PBE to exist it must be the case that e H ≤
4.
6
Consequently, for the proposed separating
PBE to exist it must be the case that e H ≤
4.
LOW Type’s IC conditions:
For any e 6= e H ,
e
e
2− L ≥ 2− .
1
1
Consequently, for the proposed separating
PBE to exist it must be the case that e H ≤
4.
LOW Type’s IC conditions:
For any e 6= e H ,
e
e
2− L ≥ 2− .
1
1
and
eH
eL
2−
≥ 4− .
1
1
Consequently, for the proposed separating
PBE to exist it must be the case that e H ≤
4.
LOW Type’s IC conditions:
For any e 6= e H ,
e
e
2− L ≥ 2− .
1
1
and
eH
eL
2−
≥ 4− .
1
1
The first one implies that e L = 0.
Consequently, for the proposed separating
PBE to exist it must be the case that e H ≤
4.
LOW Type’s IC conditions:
For any e 6= e H ,
e
e
2− L ≥ 2− .
1
1
and
eH
eL
2−
≥ 4− .
1
1
The first one implies that e L = 0.
Substitute, then, this into the second condition
and it implies that e H ≥ 2.
Consequently for the proposed separating
PBE to exist it must also be the case that
e H ≥ 2.
7
Consequently for the proposed separating
PBE to exist it must also be the case that
e H ≥ 2.
Pulling all this together, we have shown that
there exists a multiplicity of separating PBE.
In each such PBE, e L = 0 and e H ∈ [2, 4].
Consequently for the proposed separating
PBE to exist it must also be the case that
e H ≥ 2.
Pulling all this together, we have shown that
there exists a multiplicity of separating PBE.
In each such PBE, e L = 0 and e H ∈ [2, 4].
Pooling PBE
Can a Pooling Perfect Bayesian Equilibrium exist?
Suppose it does. Then it must be the case
that:
In a pooling PBE the two types of workers
choose the same education level: e H = e L = e∗ .
Consequently for the proposed separating
PBE to exist it must also be the case that
e H ≥ 2.
Pulling all this together, we have shown that
there exists a multiplicity of separating PBE.
In each such PBE, e L = 0 and e H ∈ [2, 4].
Pooling PBE
Can a Pooling Perfect Bayesian Equilibrium exist?
Suppose it does. Then it must be the case
that:
In a pooling PBE the two types of workers
choose the same education level: e H = e L = e∗ .
Furthermore, the posterior beliefs of employers
in such a pooling PBE will be as follows:
Apply Bayes rules (when can, following events
that have non-zero probability of occurring):
Pr(θ = 1 | e = e∗ ) = 1/3 and Pr(θ = 2 | e = e∗ ) =
2/3
8
Apply Bayes rules (when can, following events
that have non-zero probability of occurring):
Pr(θ = 1 | e = e∗ ) = 1/3 and Pr(θ = 2 | e = e∗ ) =
2/3
(This, posteriors are identical to priors).
Cannot apply Bayes rule following zero probability events — i.e., in the pooling PBE when
education level e is observed different from e∗ .
Apply Bayes rules (when can, following events
that have non-zero probability of occurring):
Pr(θ = 1 | e = e∗ ) = 1/3 and Pr(θ = 2 | e = e∗ ) =
2/3
(This, posteriors are identical to priors).
Cannot apply Bayes rule following zero probability events — i.e., in the pooling PBE when
education level e is observed different from e∗ .
Indeed, thus, for any e such that e 6= e∗ : Pr(θ =
1 | e) can be any number between zero and
one. The PBE concept does not restrict outof-equilibrium beliefs.
Apply Bayes rules (when can, following events
that have non-zero probability of occurring):
Pr(θ = 1 | e = e∗ ) = 1/3 and Pr(θ = 2 | e = e∗ ) =
2/3
(This, posteriors are identical to priors).
Cannot apply Bayes rule following zero probability events — i.e., in the pooling PBE when
education level e is observed different from e∗ .
Indeed, thus, for any e such that e 6= e∗ : Pr(θ =
1 | e) can be any number between zero and
one. The PBE concept does not restrict outof-equilibrium beliefs.
Suppose, then, (to most easily see whether a
pooling PBE exists), assume: Pr(θ = 1 | e) = 1
for any e such as e 6= e∗ .
That is: we assume that when employers observe education e 6= e∗ , they believe worker is
Low type for sure.
9
Suppose, then, (to most easily see whether a
pooling PBE exists), assume: Pr(θ = 1 | e) = 1
for any e such as e 6= e∗ .
That is: we assume that when employers observe education e 6= e∗ , they believe worker is
Low type for sure.
Given the above, the wages in this PBE must
be as follows (since assumed above wages equal
expected productivity):
1
2
10
∗
w(e = e ) = [2][1] + [2][2] = .
3
3
3
Suppose, then, (to most easily see whether a
pooling PBE exists), assume: Pr(θ = 1 | e) = 1
for any e such as e 6= e∗ .
That is: we assume that when employers observe education e 6= e∗ , they believe worker is
Low type for sure.
Given the above, the wages in this PBE must
be as follows (since assumed above wages equal
expected productivity):
1
2
10
∗
w(e = e ) = [2][1] + [2][2] = .
3
3
And for any e 6= e∗ , w(e) = 2(1) = 2.
3
Suppose, then, (to most easily see whether a
pooling PBE exists), assume: Pr(θ = 1 | e) = 1
for any e such as e 6= e∗ .
That is: we assume that when employers observe education e 6= e∗ , they believe worker is
Low type for sure.
Given the above, the wages in this PBE must
be as follows (since assumed above wages equal
expected productivity):
1
2
10
∗
w(e = e ) = [2][1] + [2][2] = .
3
3
3
And for any e 6= e∗ , w(e) = 2(1) = 2.
HIGH-type Incentive-Compatibility Condition is:
For any e 6= e∗ ,
10 e∗
e
− ≥ 2− .
3
2
2
10
For any e 6= e∗ ,
10 e∗
e
− ≥ 2− .
3
2
2
This is iff
10 e∗
− ≥ 2.
3
2
For any e 6= e∗ ,
10 e∗
e
− ≥ 2− .
3
2
2
This is iff
10 e∗
− ≥ 2.
3
2
Thus, for the pooling PBE to exist it must
be the case that e∗ ≤ 38 .
LOW-type Incentive-Compatibility Condition is:
For any e 6= e∗ ,
10 e∗
e
− ≥ 2− .
3
1
1
For any e 6= e∗ ,
10 e∗
e
− ≥ 2− .
3
2
2
This is iff
10 e∗
− ≥ 2.
3
2
Thus, for the pooling PBE to exist it must
be the case that e∗ ≤ 38 .
LOW-type Incentive-Compatibility Condition is:
For any e 6= e∗ ,
10 e∗
e
− ≥ 2− .
3
1
1
This is iff
10 e∗
− ≥ 2.
3
1
11
This is iff
10 e∗
− ≥ 2.
3
1
Thus, for the pooling PBE to exist it must
also be the case that e∗ ≤ 43 .
Pulling all this together implies that there exists a multiplicity pooling PBE. In each PBE,
e H = e L = e∗ ≤ 34 .