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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.