MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Risk Prerequisites July 2015 1 Introduction A key aspect of hidden information Information relates to personal characteristics ID: 469225
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Slide1
Signalling
MICROECONOMICSPrinciples and Analysis Frank Cowell
Almost essential Risk
Prerequisites
April 2018
1Slide2
Introduction
A key aspect of hidden informationInformation relates to personal characteristicshidden information about
actions is dealt with under “moral hazard”But a fundamental difference from screeninginformed party moves firstopposite case (where uninformed party moves first) dealt with under “adverse selection”Nature of strategic problemuncertainty about characteristics: game of imperfect information
updating by uninformed party in the light of the signalequilibrium concept: perfect Bayesian Equilibrium (PBE)
April 2018
2Slide3
Signalling
Agent with the information makes first move:subtly different from other “screening” problemsmove involves making a signal
Types of signalcould be a costly action (physical investment, advertising, acquiring an educational certificate) could be a costless message (manufacturers' assurances of quality, promises by service deliverers) Message is about a characteristicthis characteristic cannot be costlessly observed by others
let us call it “talent”April 2018
3Slide4
Talent
Suppose individuals differ in terms of hidden talent τ Talent is valuable in the market but possessor of τ cannot convince buyers in the marketwithout providing a signal that he has it If a signal is not possiblemay be no market equilibrium
If a signal is possiblewill there be equilibrium?more than one equilibrium?April 2018
4Slide5
Overview
Costly signals: model
Costly signals: equilibrium
Costless signals
Signalling
An educational analogy
April 2018
5Slide6
Costly signals
Suppose that a “signal” costs something physical investmentforgone incomeConsider a simple model of the labour marketSuppose productivity depends on ability
ability is not observableTwo types of workers:the able – ta the basic – tb
ta > tb Single type of jobemployers know the true product of a type t
-personif they can identify which is whichHow can able workers distinguish themselves from others?
April 2018
6Slide7
Signals: educational “investment”
Consider the decision about whether acquire educationSuppose talent on the job identical to talent at achieving educational credentials
assumed to be common knowledgemay be worth “investing” in the acquisition of credentials Education does not enhance productive abilitysimply an informative message or credentialflags up innate talent
high ability people acquire education with less effortEducation is observable certificates can be verified costlesslyfirms may use workers'’ education as an informative signal
April 2018
7Slide8
Signalling by workers
0
[LOW]
[HIGH]
1-p
p
[NOT INVEST]
[INVEST]
[NOT INVEST]
[INVEST]
f
2
[low]
[high]
[low]
[high]
[low]
[high]
[low]
[high]
f
1
[low]
[high]
[low]
[high]
[accept 2]
[reject]
[accept 1]
h
… … …
“Nature” determines worker’s type
Workers decide on education
Firms make wage offers
Workers decide whether to accept
Examine stages 1-3 more closely
investment involves time and money
simultaneous offers: Bertrand competition
h
h
April 2018
8Slide9
A model of costly signals
Previous sketch of problem is simplifiedworkers only make binary decisions (whether or not to invest)firms only make binary decisions (high or low wage)Suppose decision involve choices of z from a continuum
Ability is indexed by a person’s type tCost of acquiring education level z is C(z, t) ≥ 0C(0, t) = 0 C
z(z, t) > 0Czz(z, t) > 0 Cz
t(z, t) < 0Able person has lower cost for a given education levelAble person has lower MC for a given education level
Illustrate this for the two-type case
April 2018
9Slide10
Costly signals
0
z
C
C
(
•,
t
b
)
C
(
•,
t
a
)
z
0
C
(
z
0
,
t
a
)
C
(
z
0
,
t
b
)
(education, cost)-space
Cost function for an a type
Cost function for a b type
Costs of investment
z
0
MC of investment
z
0
April 2018
10Slide11
Payoffs to individuals
Talent does not enter the utility function directlyindividuals only care about income measure utility directly in terms of income:
v(y, z; t) := y C(
z, t)v depends on τ because talent reduces the cost of net incomeShape of C means that ICs in (z
, y)-space satisfy single-crossing: IC for a person with talent
t
is:
y
=
u
+
C
(
z
,
t
)
slope
of IC for this type is:
d
y
/
d
z
=
C
z(
z, t) for person with higher talent (
t'>t) slope of IC is: dy/dz =
Cz(z, t')
but Czt(z
, t) < 0 so IC(t') is flatter than IC(t
) at any value of z so, if IC(t') and IC(
t
) intersect at (
z
0
,
y
0
)
IC(
t
') lies above original IC(
t
) for
z
<
z
0
and below IC(
t
) for
z
>
z
1
This is important to simplify the structure of the problem
Example
y
z
high
t
low
t
0
2
4
6
8
10
12
14
16
18
0
0.5
1
1.5
2
2.5
3
3.5
C
(
z
,
t
) = (1/
t
)
z
2
April 2018
11Slide12
Rational behaviour
Workers: assume income y is determined by wageWage is conditioned on “signal” that they provide
through acquisition of educational credentialsType-τ worker chooses z to maximisew(z) C(z,
t) where w(⋅) is wage schedule that workers anticipate will be offered by firmsFirms:assume profits determined by workers’ talent
Need to design w(⋅) to max profitsdepends on beliefs about distribution of talentsconditional on value of observed signal
What will equilibrium be?
April 2018
12Slide13
Overview
Costly signals: model
Costly signals: equilibrium
Costless signals
Signalling
Costly signals discriminate among agents
Separating equilibrium
Out-of-equilibrium behaviour
Pooling equilibrium
April 2018
13Slide14
Separating equilibrium (1)
Start with a separating Perfect Bayesian EquilibriumBoth type-a and type-b agents are maximising so neither wants to switch to using the other's signalTherefore, for the talented
a-types we havef(ta) C(za, ta) ≥
f(tb) C(zb, ta)if correctly identified, no worse than if misidentified as a
b-typeLikewise for the b-types:f(t
a
)
C
(
z
a
,
t
b
) ≤
f
(
t
b
)
C
(
z
b
,
tb)Rearranging this we have C(za, t
b) C(zb, t
b) ≥ f(ta) f(
tb) positive because f(⋅) is strictly increasing and t
a > tb but since C
z > 0 this is true if and only if za > zb So able individuals acquire more education than the others
April 2018
14Slide15
Separating equilibrium (2)
If there are just two types, at the optimum zb = 0everyone knows there are only two productivity types
education does not enhance productivityso no gain to b-types in buying education So, conditions for separating equilibrium becomeC(za,
ta) ≤ f(ta) f(
tb)C(za
,
t
b
) ≥
f
(
t
a
)
f
(
t
b
)
Let
z
0
,
z
1
be the critical
z-values that satisfy these conditions with equalityz
0 such that f(tb) =
f(ta) C(z0
, tb)z1 such that
f(tb) = f(
ta) C(z1,
ta) Values z0,
z
1
set limits to education in equilibrium
remember that
C
(0,
t
)=0
April 2018
15Slide16
0
z
y
v
(
•,
t
b
)
z
0
v
(
•,
t
a
)
z
1
f
(
t
a
)
f
(
t
b
)
Bounds to education
IC for an a type
IC for a b type
critical value for a b type
critical value for an a type
both curves pass through
(0,
f
(
t
b
))
possible equilibrium
z
-values
f
(
t
a
) =
f
(
t
b
)
C
(
z
1
,
t
a
)
f
(
t
a
) =
f
(
t
b
)
C
(
z
0
,
t
b
)
Separating
eqm
: Two examples
April 2018
16Slide17
Separating equilibrium: example 1
0
v
(
•,
t
b
)
z
a
f
(
t
a
)
v
(
•,
t
a
)
w
(
•
)
“bounding” ICs for each type
wage schedule
max type-b’s utility
max type-a’s utility
•
f
(
t
b
)
•
possible equilibrium
z
-values
both curves pass through
(0,
f
(
t
b
)
)
determines
z
0
, z
1
as before
low talent acquires zero education
z
y
high talent acquires education close to
z
0
April 2018
17Slide18
Separating equilibrium: example 2
0
v
(
•,
t
b
)
f
(
t
a
)
v
(
•,
t
a
)
w
(
•
)
a
different
wage
schedule
max
type-b’s
utility
max
type-a’s
utility
f
(
t
b
)
possible
equilibrium
z
-values
just as before
low talent acquires zero education (just as before)
z
y
high talent acquires education close to
z
1
z
a
•
•
April 2018
18Slide19
Overview
Costly signals: model
Costly signals: equilibrium
Costless signals
Signalling
More on beliefs
Separating equilibrium
Out-of-equilibrium behaviour
Pooling equilibrium
April 2018
19Slide20
Out-of-equilibrium-beliefs: problem
For a given equilibrium can redraw w(⋅)-scheduleresulting attainable set for the workers must induce them to choose (za,
f(ta)) and (0, f(tb)) Shape of the w(⋅)-schedule at other values of z
? captures firms' beliefs about workers’ types in situations that do not show up in equilibriumPBE leaves open what out-of-equilibrium beliefs may beApril 2018
20Slide21
Perfect Bayesian Equilibria
Requirements for PBE do not help us to select among the separating equilibriatry common sense? Education level z0 is the minimum-cost signal for
a-types a-type's payoff is strictly decreasing in za over [z0, z1]any equilibrium with z
a > z0 is dominated by equilibrium at z0Are Pareto-dominated equilibria uninteresting?important cases of strategic interaction that produce Pareto-dominated outcomesneed a proper argument, based on the reasonableness of such an equilibrium
April 2018
21Slide22
Out-of-equilibrium beliefs: a criterion
Is an equilibrium at za > z0 “reasonable”?
requires w(•) that sets w(z′) < f(ta) for z0
< z′ < zaso firms must be assigning the belief π(z′) > 0 Imagine someone observed choosing z′b
-type IC through (z′, f(ta)
) lies below the IC through (0,
f
(
t
b
))
a
b
-type knows he’s worse off than in the separating equilibrium
a
b
-type would never go to (
z
′,
f
(
t
a
)
)
so anyone at
z
′ out of equilibrium must be an a-type An
intuitive criterion: π(z′) = 0 for any z′ (z
0, za)So only separating equilibrium worth considering is wherea-types are at (z
0, f(ta))
b-types are at (0, f(tb))
April 2018
22Slide23
Overview
Costly signals: model
Costly signals: equilibrium
Costless signals
Signalling
Agents
appear
to be al the same
Separating equilibrium
Out-of-equilibrium behaviour
Pooling equilibrium
April 2018
23Slide24
Pooling
There may be equilibria where the educational signal does not workno-one finds it profitable to "invest" in education?or all types purchase the same z
?depends on distribution of t and relationship between marginal productivity and t All workers present themselves with the same credentialsso they are indistinguishable
firms have no information to update their beliefs Firms’ beliefs are derived from the distribution of t in the populationthis distribution is common knowledge So wage offered is expected marginal productivity
E f(
t
)
:=[1
p
]
f
(
t
a
) +
p
f
(
t
b
)
Being paid this wage might be in interests of
all
workers
Example
April 2018
24Slide25
0
z
y
v
(
•,
t
b
)
z
0
v
(
•,
t
a
)
z
1
f
(
t
a
)
f
(
t
b
)
E
f
(
t
)
No signals: an example
possible
z-
values with signalling
outcome under signalling
outcome without signalling
•
highest a-type IC under signalling
both pass through
(0,
E
f
(
t
)
)
the type-b IC must be higher than with signalling
but,
in this case
, so is the type-a IC
z
0
should school be banned?
April 2018
25Slide26
critical
z
for b-type to accept pooling payoff
0
z
y
v
(
•,
t
b
)
z
2
f
(
t
a
)
f
(
t
b
)
E
f
(
t
)
Pooling: limits on
z
?
critical IC for a b-type
E
f
(
t
) = [1
p
]
f
(
t
a
)
+
pf
(
t
b
)
expected marginal productivity
[1
p
]
f
(
t
a
) +
pf
(
t
b
)
C
(
z
2
,
t
b
) =
f
(
t
b
)
b-type payoff with 0 education
viable
z
-values in pooling
eqm
April 2018
26Slide27
Pooling equilibrium: example 1
0
z
y
v
(
•,
t
b
)
v
(
•,
t
a
)
w
(
•
)
z
*
f
(
t
a
)
f
(
t
b
)
E
f
(
t
)
expected marginal productivity
viable
z-
values in pooling
eqm
wage schedule
utility maximisation
equilibrium education
April 2018
27Slide28
Pooling equilibrium: example 2
0
z
y
v
(
•,
t
b
)
v
(
•,
t
a
)
w
(
•
)
z
*
f
(
t
a
)
f
(
t
b
)
expected marginal productivity
viable
z-
values in pooling
eqm
wage schedule
utility maximisation
equilibrium education
E
f
(
t
)
but is pooling consistent with out-of-equilibrium behaviour?
April 2018
28Slide29
0
z
y
v
(
•,
t
b
)
z
0
v
(
•,
t
a
)
f
(
t
a
)
f
(
t
b
)
E
f
(
t
)
z
'
z
*
Intuitive criterion again
a pooling equilibrium
a critical
z
-
value
z
'
E
f
(
t
)
C
(
z
*
,
t
b
) =
f
(
t
a
)
C
(
z
′,
t
b
)
wage offer for an a-type at
z
0
> z
'
max b-type utility at
z
0
max a-type utility at
z
0
b-type would not choose
z
0
under intuitive criterion
p
(
z
0
) = 0
a-type gets higher utility at
z
0
would move from
z*
to
z
0
so pooling
eqm
inconsistent with intuitive criterion
April 2018
29Slide30
Overview
Costly signals: model
Costly signals: equilibrium
Costless signals
Signalling
An argument by example
April 2018
30Slide31
Costless signals: an example
Present the issue with a simplified examplegeneral treatments can be difficultN risk-neutral agents share in a project with outputq =
a[z1×z2×z3×...] where 0 < α < 1zh = 0 or 1 is participation indicator of agent h
Agent h has cost of participation ch (unknown to others)ch [0,1]it is common knowledge that
prob(ch ≤ c) = cOutput is a public good, so net payoff to each agent
h
is
q
c
h
Consider this as a simultaneous-move game
what is the NE?
improve on NE by making announcements before the game starts?
April 2018
31Slide32
Example: NE without signals
Central problem: each h risks incurring cost ch while getting consumption 0 If π is probability that any other agent participates, payoff to
h is a −ch with probability [p]N−1−
ch otherwise Expected payoff to h is a[p]N−1 −
chProbability that expected payoff is positive is a[p
]
N
−1
but this is the probability that agent
h
actually participates
therefore
p
=
a
[
p
]
N
−1
this can only be satisfied if
p
= 0
So the NE is
z
h = 0 for all h, as long as α < 1
April 2018
32Slide33
Example: introduce signals
Introduce a preliminary stage to the gameEach agent has the opportunity to signal his intention:each agent announces [YES] or [NO] to the others
each agent then decides whether or not to participateThen there is an equilibrium in which the following occurseach h announces [YES] if and only if ch < α
h selects zh = 1 iff all agents have announced [YES]In this equilibrium:agents don’t risk wasted effort
if there are genuine high-cost ch agents present that inhibit the projectthis will be announced at the signalling stage
April 2018
33Slide34
Signalling: summary
Both costly and costless signals are important Costly signals:separating PBE not unique?intuitive criterion suggests out-of-equilibrium beliefspooling equilibrium may not be uniqueinconsistent with intuitive criterion? Costless signals:
a role to play in before the game startsApril 2018
34