Post on 18-Dec-2021
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 .
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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.
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
ce (η, e) : marginal cost of education
Assumption:ce (L, e) > ce (H, e)
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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.
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
The �rst-best (the case of observable types)
e� (η) solves maxefy(η, e)� c(η, e)g
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
Two cases are relevant:
1) the no envy case
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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.
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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 .
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
The worker�s problem:
Given the wage schedule, a worker of type η solves
maxefw(e)� c(η, e)g
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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.
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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.
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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.
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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?
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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 .
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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.
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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.
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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.
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
The following graph depicts a hybrid equilibrium:
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1
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.
Dezsö Szalay (University of Bonn) Economics of Contracts and Information 2014 1 / 1