lecture notes on spence’s job market … · lecture notes on spence’s job market signalling 1....
TRANSCRIPT
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) = 13, and Pr(θ = 2) = 2
3.
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) = 13, and Pr(θ = 2) = 2
3.
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) = 13, and Pr(θ = 2) = 2
3.
Productivity of worker is 2θ
Cost of education e is C(e) = eθ .
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) = 13, and Pr(θ = 2) = 2
3.
Productivity of worker is 2θ
Cost of education e is C(e) = eθ .
Signalling game: First, the worker chooses the
level of eduction, e. The employer, upon ob-
serving e, chooses wage.
PERFECT BAYESIAN EQUILIBRIA.
3
PERFECT BAYESIAN EQUILIBRIA.
Simplify Analysis: Assume wage equals to ex-
pected productivity.
PERFECT BAYESIAN EQUILIBRIA.
Simplify Analysis: Assume wage equals to ex-
pected 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 eH and
eL denote the levels chosen by high and low
ability types, respectively, where eH 6= eL.
PERFECT BAYESIAN EQUILIBRIA.
Simplify Analysis: Assume wage equals to ex-
pected 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 eH and
eL denote the levels chosen by high and low
ability types, respectively, where eH 6= eL.
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 = eH) = 1 and Pr(θ = 1 | e = eL) = 1
4
Apply Bayes rules (when can, following events
that have non-zero probability of occurring):
Pr(θ = 2 | e = eH) = 1 and Pr(θ = 1 | e = eL) = 1
Cannot apply Bayes rule following zero prob-
ability events — i.e., in the separating PBE
when education level e is observed different
from eH and eL.
Apply Bayes rules (when can, following events
that have non-zero probability of occurring):
Pr(θ = 2 | e = eH) = 1 and Pr(θ = 1 | e = eL) = 1
Cannot apply Bayes rule following zero prob-
ability events — i.e., in the separating PBE
when education level e is observed different
from eH and eL.
Indeed, thus, for any e such that e 6= eH and
e 6= eL: Pr(θ = 1 | e) can be any number be-
tween 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 = eH) = 1 and Pr(θ = 1 | e = eL) = 1
Cannot apply Bayes rule following zero prob-
ability events — i.e., in the separating PBE
when education level e is observed different
from eH and eL.
Indeed, thus, for any e such that e 6= eH and
e 6= eL: Pr(θ = 1 | e) can be any number be-
tween zero and one. The PBE concept does
not restrict out-of-equilibrium beliefs.
Suppose, then, (to most easily see whether aseparating PBE exists), assume: Pr(θ = 1 | e) =1 for any e such as e 6= eH and e 6= eL.
That is: we assume that when employers ob-serve education e 6= eH, they believe worker isLow type for sure.
Given the above, the wages in this PBE must
be as follows (since assumed above wages equal
expected productivity):
w(e = eH) = 2(2) = 4 and for any e 6= eH, 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 = eH) = 2(2) = 4 and for any e 6= eH, w(e) =2(1) = 2.
INCENTIVE COMPATIBILITY CONDITIONS
HIGH Type’s IC conditions:
For any e 6= eH,
4− eH2≥ 2− e
2.
Given the above, the wages in this PBE must
be as follows (since assumed above wages equal
expected productivity):
w(e = eH) = 2(2) = 4 and for any e 6= eH, w(e) =2(1) = 2.
INCENTIVE COMPATIBILITY CONDITIONS
HIGH Type’s IC conditions:
For any e 6= eH,
4− eH2≥ 2− e
2.
This implies the High type IC condition be-
comes:
4− eH2≥ 2.
Consequently, for the proposed separating
PBE to exist it must be the case that eH ≤4.
6
Consequently, for the proposed separating
PBE to exist it must be the case that eH ≤4.
LOW Type’s IC conditions:
For any e 6= eH,
2− eL1≥ 2− e
1.
Consequently, for the proposed separating
PBE to exist it must be the case that eH ≤4.
LOW Type’s IC conditions:
For any e 6= eH,
2− eL1≥ 2− e
1.
and
2− eL1≥ 4− eH
1.
Consequently, for the proposed separating
PBE to exist it must be the case that eH ≤4.
LOW Type’s IC conditions:
For any e 6= eH,
2− eL1≥ 2− e
1.
and
2− eL1≥ 4− eH
1.
The first one implies that eL = 0.
Consequently, for the proposed separating
PBE to exist it must be the case that eH ≤4.
LOW Type’s IC conditions:
For any e 6= eH,
2− eL1≥ 2− e
1.
and
2− eL1≥ 4− eH
1.
The first one implies that eL = 0.
Substitute, then, this into the second condition
and it implies that eH ≥ 2.
Consequently for the proposed separating
PBE to exist it must also be the case that
eH ≥ 2.
7
Consequently for the proposed separating
PBE to exist it must also be the case that
eH ≥ 2.
Pulling all this together, we have shown that
there exists a multiplicity of separating PBE.
In each such PBE, eL = 0 and eH ∈ [2, 4].
Consequently for the proposed separating
PBE to exist it must also be the case that
eH ≥ 2.
Pulling all this together, we have shown that
there exists a multiplicity of separating PBE.
In each such PBE, eL = 0 and eH ∈ [2, 4].
Pooling PBE
Can a Pooling Perfect Bayesian Equilibrium ex-
ist?
Suppose it does. Then it must be the case
that:
In a pooling PBE the two types of workers
choose the same education level: eH = eL = e∗.
Consequently for the proposed separating
PBE to exist it must also be the case that
eH ≥ 2.
Pulling all this together, we have shown that
there exists a multiplicity of separating PBE.
In each such PBE, eL = 0 and eH ∈ [2, 4].
Pooling PBE
Can a Pooling Perfect Bayesian Equilibrium ex-
ist?
Suppose it does. Then it must be the case
that:
In a pooling PBE the two types of workers
choose the same education level: eH = eL = e∗.
Furthermore, the posterior beliefs of employersin 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 proba-
bility 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 proba-
bility 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 out-
of-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 proba-
bility 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 out-
of-equilibrium beliefs.
Suppose, then, (to most easily see whether a
pooling PBE exists), assume: Pr(θ = 1 | e) = 1for any e such as e 6= e∗.
That is: we assume that when employers ob-
serve 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) = 1for any e such as e 6= e∗.
That is: we assume that when employers ob-
serve 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):
w(e = e∗) =13[2][1] +
23[2][2] =
103
.
Suppose, then, (to most easily see whether a
pooling PBE exists), assume: Pr(θ = 1 | e) = 1for any e such as e 6= e∗.
That is: we assume that when employers ob-
serve 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):
w(e = e∗) =13[2][1] +
23[2][2] =
103
.
And for any e 6= e∗, w(e) = 2(1) = 2.
Suppose, then, (to most easily see whether a
pooling PBE exists), assume: Pr(θ = 1 | e) = 1for any e such as e 6= e∗.
That is: we assume that when employers ob-
serve 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):
w(e = e∗) =13[2][1] +
23[2][2] =
103
.
And for any e 6= e∗, w(e) = 2(1) = 2.
HIGH-type Incentive-Compatibility Condition is:
For any e 6= e∗,103− e∗
2≥ 2− e
2.
10
For any e 6= e∗,103− e∗
2≥ 2− e
2.
This is iff
103− e∗
2≥ 2.
For any e 6= e∗,103− e∗
2≥ 2− e
2.
This is iff
103− e∗
2≥ 2.
Thus, for the pooling PBE to exist it must
be the case that e∗ ≤ 83.
LOW-type Incentive-Compatibility Condition is:
For any e 6= e∗,103− e∗
1≥ 2− e
1.
For any e 6= e∗,103− e∗
2≥ 2− e
2.
This is iff
103− e∗
2≥ 2.
Thus, for the pooling PBE to exist it must
be the case that e∗ ≤ 83.
LOW-type Incentive-Compatibility Condition is:
For any e 6= e∗,103− e∗
1≥ 2− e
1.
This is iff
103− e∗
1≥ 2.
11
This is iff
103− e∗
1≥ 2.
Thus, for the pooling PBE to exist it must
also be the case that e∗ ≤ 43.
Pulling all this together implies that there ex-
ists a multiplicity pooling PBE. In each PBE,
eH = eL = e∗ ≤ 43.