Frank Cowell: Signalling SIGNALLING MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Risk Almost essential Risk Prerequisites July

Frank Cowell: Signalling

SIGNALLINGMICROECONOMICSPrinciples and Analysis Frank Cowell

Almost essential Risk

Prerequisites

July 2015 1


Introduction A key aspect of hidden information Information relates to personal characteristics

• hidden information about actions is dealt with under “moral hazard” But a fundamental difference from screening

• informed party moves first• opposite case (where uninformed party moves first) dealt with under

“adverse selection” Nature of strategic problem

• uncertainty about characteristics: game of imperfect information• updating by uninformed party in the light of the signal• equilibrium concept: perfect Bayesian Equilibrium (PBE)

July 2015 2


Signalling Agent with the information makes first move:

• subtly different from other “screening” problems• move involves making a signal

Types of signal• could 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 characteristic

• this characteristic cannot be costlessly observed by others• let us call it “talent”

July 2015 3


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 market• without providing a signal that he has it

If a signal is not possible• may be no market equilibrium

If a signal is possible• will there be equilibrium?• more than one equilibrium?

July 2015 4


Overview

Costly signals: model

Costly signals: equilibrium

Costless signals

Signalling

An educational analogy

July 2015 5


Costly signals Suppose that a “signal” costs something

• physical investment• forgone income

Consider a simple model of the labour market Suppose productivity depends on ability

• ability is not observable Two types of workers:

• the able – ta • the basic – tb

• ta > tb Single type of job

• employers know the true product of a type t-person• if they can identify which is which

How can able workers distinguish themselves from others?

July 2015 6


Signals: educational “investment” Consider the decision about whether acquire education Suppose talent on the job identical to talent at achieving

educational credentials • assumed to be common knowledge• may be worth “investing” in the acquisition of credentials

Education does not enhance productive ability• simply an informative message or credential• flags up innate talent• high ability people acquire education with less effort

Education is observable • certificates can be verified costlessly• firms may use workers'’ education as an informative signal

July 2015 7


Signalling by workers0

[LOW] [HIGH]1-pp

[NOT INVEST]

[INVEST][NOT INVEST]

[INVEST]

f2

[low][high]

[low][high]

[low][high]

[low][high]

f1

[low] [high][low] [high]

[accept 2][rejec

t]

[acc

ept 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

July 2015 8


A model of costly signals Previous sketch of problem is simplified

• workers 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 t Cost of acquiring education level z is C(z, t) ≥ 0

• C(0, t) = 0 Cz(z, t) > 0• Czz(z, t) > 0 Czt(z, t) < 0

Able person has lower cost for a given education level Able person has lower MC for a given education level Illustrate this for the two-type case

July 2015 9


Costly signals

0

z

C

C(•,tb)

C(•,ta)

z0

C(z0,ta)

C(z0,tb)

(education, cost)-space Cost function for an a type Cost function for a b type Costs of investment z0 MC of investment z0

July 2015 10


Payoffs to individuals Talent does not enter the utility function directly

• individuals 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 income

Shape of C means that ICs in (z, y)-space satisfy single-crossing condition: • IC for a person with talent t is: y = u + C(z, t) • slope of IC for this type is: dy/dz = Cz(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 (z0, y0) • IC(t') lies above original IC(t) for z < z0 and below IC(t) for z > z1

This is important to simplify the structure of the problemExample

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) z2

July 2015 11


Rational behaviourWorkers:

• assume income y is determined by wage Wage is conditioned on “signal” that they provide

• through acquisition of educational credentials Type-τ worker chooses z to maximise

• w(z) - C(z, t) • where w( ) is the wage schedule that workers anticipate will be offered by ⋅

firms Firms:

• assume profits determined by workers’ talent Need to design w( ) to max profits⋅

• depends on beliefs about distribution of talents• conditional on value of observed signal

What will equilibrium be?

July 2015 12


Overview



Costless signals

Signalling

Costly signals discriminate among agents

• Separating equilibrium• Out-of-equilibrium behaviour• Pooling equilibrium

July 2015 13


Separating equilibrium (1) Start with a separating Perfect Bayesian Equilibrium Both type-a and type-b agents are maximising

• so neither wants to switch to using the other's signal Therefore, for the talented a-types we have

• f(ta) - C(za, ta) ≥ f(tb) - C(zb, ta)• if correctly identified, no worse than if misidentified as a b-type

Likewise for the b-types:• f(ta) - C(za, tb) ≤ f(tb) - C(zb, tb)

Rearranging this we have• C(za, tb) - C(zb, tb) ≥ f(ta) - f(tb) • positive because f( ) is strictly increasing and ⋅ ta > tb • but since Cz > 0 this is true if and only if za > zb

So able individuals acquire more education than the others

July 2015 14


Separating equilibrium (2) If there are just two types, at the optimum zb = 0

• everyone knows there are only two productivity types• education does not enhance productivity• so no gain to b-types in buying education

So, conditions for separating equilibrium become• C(za, ta) ≤ f(ta) - f(tb)• C(za, tb) ≥ f(ta) - f(tb)

Let z0, z1 be the critical z-values that satisfy these conditions with equality• z0 such that f(tb) = f(ta) - C(z0, tb)• z1 such that f(tb) = f(ta) - C(z1, ta)

Values z0, z1 set limits to education in equilibrium

remember that C(0, t)=0

July 2015 15


0 z

y

v(•,tb)

z0

v(•,ta)

z1

f(ta)

f(tb)

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(tb))

possible equilibrium z-values

f(ta) = f (tb) - C(z1, ta) f(ta) = f (tb) - C(z0, tb)

Separating eqm: Two examples

July 2015 16


Separating equilibrium: example 1

0

v(•,tb)

za

f(ta)

v(•,ta)

w(•)

“bounding” ICs for each type

wage schedule max type-b’s utility max type-a’s utility

•

f(tb) •


both curves pass through (0, f(tb))

determines z0, z1 as before

low talent acquires zero education

z

y

high talent acquires education close to z0

July 2015 17


Separating equilibrium: example 2

0

v(•,tb)

f(ta)

v(•,ta)

w(•)

a different wage schedule max type-b’s utility max type-a’s utility

f(tb)


just as before low talent acquires zero

education (just as before)

z

y

high talent acquires education close to z1

za

•

•

July 2015 18


Overview



Costless signals

Signalling

More on beliefs• Separating equilibrium• Out-of-equilibrium behaviour• Pooling equilibrium

July 2015 19


Out-of-equilibrium-beliefs: problemFor a given equilibrium can redraw w( )-schedule⋅

• resulting 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

be

July 2015 20


Perfect Bayesian Equilibria Requirements for PBE do not help us to select among the

separating equilibria• try 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 za > z0 is dominated by equilibrium at z0

Are Pareto-dominated equilibria uninteresting?• important cases of strategic interaction that produce Pareto-dominated

outcomes• need a proper argument, based on the reasonableness of such an

equilibrium

July 2015 21


Out-of-equilibrium beliefs: a criterion Is an equilibrium at za > z0 “reasonable”?

• requires w(•) that sets w(z′) < f(ta) for z0 < z′ < za

• so 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(tb))• a b-type knows he’s worse off than in the separating equilibrium• a b-type would never go to (z′, f(ta)) • so anyone at z′ out of equilibrium must be an a-type

An intuitive criterion: • π(z′) = 0 for any z′ (z0, za)

So only separating equilibrium worth considering is where• a-types are at (z0, f(ta)) • b-types are at (0, f(tb))

July 2015 22


Overview



Costless signals

Signalling

Agents appear to be al the same • Separating equilibrium

• Out-of-equilibrium behaviour• Pooling equilibrium

July 2015 23


Pooling There may be equilibria where the educational signal does not work

• no-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 credentials• so they are indistinguishable• firms have no information to update their beliefs

Firms’ beliefs are derived from the distribution of t in the population• this distribution is common knowledge

So wage offered is expected marginal productivity• E f(t):=[1 - p]f(ta) + pf(tb)

Being paid this wage might be in interests of all workers Example

July 2015 24


0z

y

v(•,tb)

z0

v(•,ta)

z1

f(ta)

f(tb)

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

z0

should school be banned?

July 2015 25


critical z for b-type to accept pooling payoff

0z

y

v(•,tb)

z2

f(ta)

f(tb)

E f(t)

Pooling: limits on z? critical IC for a b-type

E f(t) = [1-p]f(ta) + pf(tb)

expected marginal productivity

[1-p] f(ta) + pf(tb) - C(z2, tb) = f(tb)

b-type payoff with 0 education

viable z -values in pooling eqm

July 2015 26


Pooling equilibrium: example 1

0 z

y v(•,tb) v(•,ta)

w(•)

z*

f(ta)

f(tb)

E f(t)

expected marginal productivity viable z-values in pooling eqm wage schedule utility maximisation equilibrium education

July 2015 27


Pooling equilibrium: example 2

0 z

y v(•,tb) v(•,ta)

w(•)

z*

f(ta)

f(tb)

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?

July 2015 28


0

z

y

v(•,tb)

z0

v(•,ta)

f(ta)

f(tb)

E f(t)

z'z*

Intuitive criterion again a pooling equilibrium a critical z-value z'

E f(t) - C(z*, tb) = f(ta) - C(z′,tb)

wage offer for an a-type at z0 > z' max b-type utility at z0

max a-type utility at z0

b-type would not choose z0 under intuitive criterion p(z0) = 0 a-type gets higher utility at z0 would move from z* to z0 so pooling eqm inconsistent

with intuitive criterion

July 2015 29


Overview



Costless signals

Signalling

An argument by example

July 2015 30


Costless signals: an example Present the issue with a simplified example

• general treatments can be difficult N risk-neutral agents share in a project with output

• q = a[z1×z2×z3×...] where 0 < α < 1• zh = 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) = c

Output is a public good, so net payoff to each agent h is • q - ch

Consider this as a simultaneous-move game• what is the NE?• improve on NE by making announcements before the game starts?

July 2015 31


Example: NE without signals Central problem: each h risks incurring cost ch while getting

consumption 0 If π is the 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 − ch

Probability 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 zh = 0 for all h, as long as α < 1

July 2015 32


Example: introduce signals Introduce a preliminary stage to the game Each 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 participate

Then there is an equilibrium in which the following occurs• each 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 project• this will be announced at the signalling stage

July 2015 33


Signalling: summary Both costly and costless signals are important Costly signals:

• separating PBE not unique?• intuitive criterion suggests out-of-equilibrium beliefs• pooling equilibrium may not be unique• inconsistent with intuitive criterion?

Costless signals:• a role to play in before the game starts

July 2015 34

Documents

Frank Cowell: Signalling SIGNALLING MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Risk Almost essential Risk Prerequisites July