Economics of Contracts and Information
Transcript of 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 .
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