Economics of Contracts and Information

Economics of Contracts and Information

Dezsö Szalay

University of Bonn

2014

Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1

Economics of Contracts and InformationUniversity of Bonn

Professor Dezsö Szalay2014

3. Job Market Signaling (Spence (1973))This material is based on Gibbons, R. (1992) A primer in Game Theory,chapter 4

Nature determines a worker�s type η 2 fH, Lg .prob (η = H) = q.

Worker learns ability and then chooses education e � 0.Two �rms observe education (but not type) and then makesimultaneous wage o¤ers.

The worker accepts the highest o¤er, if any, if indi¤erent, acceptseach will probability 1

2 .


Payo¤s:

Workerw � c(η, e)

Firmy(η, e)� w if o¤er accepted

0 otherwise.

Di¤erences in e are supposed to be di¤erences in performance, not in thesense of the duration of education; so the number and kind of coursestaken would be OK.


ce (η, e) : marginal cost of education

Assumption:ce (L, e) > ce (H, e)


Indi¤erence curves

Worker of type L requires a higher increase in wage to compensate him foran increase in e.Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1

Competition between �rms implies

w(e) = µ(H, e) � y(H, e) + (1� µ(H, e)) � y(L, e)where

µ(H, e) = market�s belief that worker is of type H

Assumption: Firms�beliefs are the same on and o¤ equilibrium path

On equilibrium path, this is true by de�nition of equilibrium; o¤equilibrium path, this amounts to an assumption.


The �rst-best (the case of observable types)

e� (η) solves maxefy(η, e)� c(η, e)g


Two cases are relevant:

1) the no envy case


2) the envy case

Type L envies Type H.Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1

The situation depicted above cannot be an equilibrium with unobservableworker types. Why?

What can be an equilibrium?

Pooling; Separating, Hybrid equilibrium


A pooling equilibrium

In a pooling equilibrium all types behave the same way; hence, there isnothing to be learned from this behavior; posterior beliefs are equal toprior beliefs.Given prior beliefs, �rms o¤er the following wage schedule:

wp = q � y(H, ep) + (1� q) � y(L, ep)

beliefs on path are determined by prior.

To construct the equilibrium, we need to determine beliefs o¤ path,that is, for e 6= ep .Beliefs in turn determine the �rm�s strategy o¤ path.


One possibility:

µ(H, e) =�0 for e 6= epq for e = ep

Consistently with these beliefs, �rms o¤er the following wage schedule:

w(e) =�y (L, e) for e 6= epwp for e = ep .


The worker�s problem:

Given the wage schedule, a worker of type η solves

maxefw(e)� c(η, e)g


No worker type has any incentive to deviate:

In particular, the low type has no incentive to deviate as long as ep � e 0.

The high type has no incentive to deviate.


Other pooling equilibria

Equilibria may di¤er on equilibrium path; that is, they may induce adi¤erent level of e.

In particular, some level of e � e 0 will do.

Equilibria may di¤er o¤ path; that is, they may specify di¤erent beliefs o¤path.


A pooling equilibrium with di¤erent o¤ path beliefs:

µ(H j e) =

8<:0 for e � e 00 except for e = epq for e = epq for e > e 00.

w(e) =

8<:y(L, e) for e � e 00 except for e = epwp for e = epwp for e > e 00.

If e� (L) � ep � e 0, low type has no incentive to deviate.Since e > e 00 is seen as average type, high type has no incentive to deviate.


Seperating equilibrium

No envy-case is not interesting. Consider thus the envy case.

Immediate insight: in any seperating equilibrium, the lowest typemust exert e¤ort e�(L)

What ist the e¤ort level of the high type?


Wage functions:

w(e) =�y(L, e) for e < esy(H, e) for e � es .

associated beliefs:

µ(H j e) =�0 for e < es1 for e � es .


Other seperating equilibria

Equilibria may di¤er with respect to on path behavior (that is, a di¤erentlevel of es );as long as es is not too high, this is still an equilibrium.

Equilibria may di¤er with respect to beliefs o¤ path. Consider, e.g., thefollowing beliefs:

µ(H j e) =

8<:0 for e � e�(H)ε for e�(H) < e < es1 for e � es .

The wage function is adjusted accordingly.For ε su¢ ciently small, this is an equilibrium.


Hybrid equilibriaConsider an equilibrium where the low type mixes between eh and e�(L).By the same reasoning as above, e�(L) must be in the support of themixed strategy.Beliefs are then for e = eh :

µ(H j eh) =q

q + (1� q)π ,

where π is the probability with which the low type chooses eh.


The following wage schedule is optimal against these beliefs:

wh =q

q + (1� q)π| {z }�r

� y(H, eh) +(1� q)π

q + (1� q)π| {z }�1�r

� y(L, eh).

The wage schedule wh is above the wage schedule with prior beliefs.This is true since

q < r () q + (1� q)π < 1() π < 1.


The following graph depicts a hybrid equilibrium:


Beliefs are

µ(H j e) =�0 for e < ehr for e � eh.

and the wage schedule which is optimal against these beliefs is

w(e) =�

y(L, e) for e < ehr � y(H, e) + (1� r) � y(L, e) for e � eh.


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