2. Frank Cowell: Microeconomics Signalling MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Risk Almost essential Risk Prerequisites February 2007

3. Frank Cowell: Microeconomics Introduction A key aspect of hidden information A key aspect of hidden information Information relates to personal characteristics Information relates to personal characteristics  hidden information about actions is dealt with under “moral hazard” But a fundamental difference from screening 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 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) Jump to “Moral Hazard” Jump to “Adverse selection”

4. Frank Cowell: Microeconomics Signalling Agent with the information makes first move: Agent with the information makes first move:  subtly different from other “screening” problems  move involves making a signal Types of 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 Message is about a characteristic  this characteristic cannot be costlessly observed by others  let us call it “talent”…

5. Frank Cowell: Microeconomics Talent Suppose individuals differ in terms of hidden talent τ Suppose individuals differ in terms of hidden talent τ Talent is valuable in the market 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 If a signal is not possible  may be no market equilibrium If a signal is possible If a signal is possible  will there be equilibrium?  …more than one equilibrium?

6. Frank Cowell: Microeconomics Overview... Costly signals: model Costly signals: equilibrium Costless signals Signalling An educational analogy

7. Frank Cowell: Microeconomics Costly signals Suppose that a “signal” costs something Suppose that a “signal” costs something  physical investment…  forgone income… Consider a simple model of the labour market Consider a simple model of the labour market Suppose productivity depends on ability Suppose productivity depends on ability  Ability is not observable Two types of workers: Two types of workers:  the able –  a  the bog-standard –  b   a >  b Single type of job Single type of job  employers know the true product of a type  -person…  …if they can identify which is which How can able workers distinguish themselves from others? How can able workers distinguish themselves from others?

8. Frank Cowell: Microeconomics Signals: educational “investment” Consider the decision about whether acquire education Consider the decision about whether acquire education Suppose talent on the job identical to talent at achieving educational credentials 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 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 Education is observable  certificates can be verified costlessly  firms may use workers'’ education as an informative signal

9. Frank Cowell: Microeconomics Signalling by workers 0 [LOW][HIGH]  [NOT INVEST] [INVEST] [NOT INVEST] [INVEST] f2f2 [low] [high] [low] [high] [low] [high] [low] [high] f1f1 [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 hh

10. Frank Cowell: Microeconomics A model of costly signals Previous sketch of problem is simplified 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 Suppose decision involve choices of z from a continuum Ability is indexed by a person’s type  Ability is indexed by a person’s type  Cost of acquiring education level z is C(z,  ) ≥ 0 Cost of acquiring education level z is C(z,  ) ≥ 0  C(0,  ) = 0C z (z,  ) > 0  C zz (z,  ) > 0C z  (z,  ) 0C z  (z,  ) < 0 Able person has lower cost for a given education level Able person has lower cost for a given education level Able person has lower MC for a given education level Able person has lower MC for a given education level Illustrate this for the two-type case Illustrate this for the two-type case

11. Frank Cowell: Microeconomics Costly signals 0 z C C(,b)C(,b) C(,a)C(,a) z0z0 C(z0,a)C(z0,a) C(z0,b)C(z0,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

12. Frank Cowell: Microeconomics Payoffs to individuals Talent does not enter the worker's utility function directly Talent does not enter the worker's utility function directly  individuals only care about income  measure utility directly in terms of income:  v(y, z;  ) := y  C(z,  )  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: Shape of C means that ICs in (z, y)-space satisfy single-crossing condition:  IC for a person with talent  is: y =  + C(z,  )  slope of IC for this type is: dy/dz = C z (z,  )  for person with higher talent (  '>  ) slope of IC is: dy/dz = C z (z,  ')  but C z  (z,  ) < 0 so IC(  ') is flatter than IC(  ) at any value of z  so, if IC(  ') and IC(  ) intersect at (z 0, y 0 )…  IC(  ') lies above original IC(  ) for z z 1 This is important to simplify the structure of the problem This is important to simplify the structure of the problem Example y z high  low  0 2 4 6 8 10 12 14 16 18 00.511.522.533.5 C(z,  ) = (1/  ) z 2

13. Frank Cowell: Microeconomics Rational behaviour Workers: Workers:  assume income y is determined by wage Wage is conditioned on “signal” that they provide Wage is conditioned on “signal” that they provide  through acquisition of educational credentials Type-τ worker chooses z to maximise Type-τ worker chooses z to maximise  w(z)  C(z,  )  where w( ⋅ ) is the wage schedule that workers anticipate will be offered by firms Firms: Firms:  assume profits determined by workers’ talent Need to design w( ⋅ ) to max profits Need to design w( ⋅ ) to max profits  depends on beliefs about distribution of talents.. ..conditional on value of observed signal What will equilibrium be? What will equilibrium be?

14. Frank Cowell: Microeconomics Overview... Costly signals: model Costly signals: equilibrium Costless signals Signalling Costly signals discriminate among agents Separating equilibrium Out-of-equilibrium behaviour Pooling equilibrium

15. Frank Cowell: Microeconomics Separating equilibrium (1) Start with a separating Perfect Bayesian Equilibrium Start with a separating Perfect Bayesian Equilibrium Both type-a and type-b agents are maximising 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 Therefore, for the talented a-types we have   (  a )  C(z a,  a ) ≥  (  b )  C(z b,  a )  if correctly identified, no worse than if misidentified as a b-type Likewise for the b-types: Likewise for the b-types:   (  a )  C(z a,  b ) ≤  (  b )  C(z b,  b ) Rearranging this we have Rearranging this we have  C(z a,  b )  C(z b,  b ) ≥  (  a )   (  b )  positive because  ( ⋅ ) is strictly increasing and  a >  b  but since C z > 0 this is true if and only if z a > z b So able individuals acquire more education than the others So able individuals acquire more education than the others

16. Frank Cowell: Microeconomics Separating equilibrium (2) If there are just two types, at the optimum z b = 0 If there are just two types, at the optimum z b = 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 So, conditions for separating equilibrium become  C(z a,  a ) ≤  (  a )   (  b )  C(z a,  b ) ≥  (  a )   (  b ) Let z 0, z 1 be the critical z-values that satisfy these conditions with equality Let z 0, z 1 be the critical z-values that satisfy these conditions with equality  z 0 such that  (  b ) =  (  a )  C(z 0,  b )  z 1 such that  (  b ) =  (  a )  C(z 1,  a ) Values z 0, z 1 set limits to education in equilibrium… Values z 0, z 1 set limits to education in equilibrium… remember that C(0,  )=0

17. Frank Cowell: Microeconomics 0 z y v(,b)v(,b) z0z0 v(,a)v(,a) z1z1 (a)(a) (b)(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, )  both curves pass through (0,  (  b ))   possible equilibrium z-values   (  a ) =  (  b )  C(z 1,  a )   (  a ) =  (  b )  C(z 0,  b ) Separating eqm: Two examples

18. Frank Cowell: Microeconomics Separating equilibrium: example 1 0 v(,b)v(,b) zaza (a)(a) v(,a)v(,a) w()   “bounding” ICs for each type   wage schedule   max type-b’s utility   max type-a’s utility (b)(b)   possible equilibrium z-values  both curves pass through (0, )  both curves pass through (0,  (  b ))  determines z 0, z 1 as before  low talent acquires zero education z y  high talent acquires education close to z 0

19. Frank Cowell: Microeconomics Separating equilibrium: example 2 0 v(,b)v(,b) (a)(a) v(,a)v(,a) w()   a different wage schedule   max type-b’s utility   max type-a’s utility (b)(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 zaza

20. Frank Cowell: Microeconomics Overview... Costly signals: model Costly signals: equilibrium Costless signals Signalling More on beliefs… Separating equilibrium Out-of-equilibrium behaviour Pooling equilibrium

21. Frank Cowell: Microeconomics Out-of-equilibrium-beliefs: problem For a given equilibrium can redraw w( ⋅ )-schedule For a given equilibrium can redraw w( ⋅ )-schedule  resulting attainable set for the workers must induce them to choose (z a,  (  a )) and (0,  (  b )) Shape of the w( ⋅ )-schedule at other values of z? Shape of the w( ⋅ )-schedule at other values of z?  captures firms' beliefs about workers’ types in situations that do not show up in equilibrium PBE leaves open what out-of-equilibrium beliefs may be PBE leaves open what out-of-equilibrium beliefs may be

22. Frank Cowell: Microeconomics Perfect Bayesian Equilibria Requirements for PBE do not help us to select among the separating equilibria Requirements for PBE do not help us to select among the separating equilibria  try common sense? Education level z 0 is the minimum-cost signal for a-types Education level z 0 is the minimum-cost signal for a-types  a-type's payoff is strictly decreasing in z a over [z 0, z 1 ]  any equilibrium with z a > z 0 is dominated by equilibrium at z 0 Are Pareto-dominated equilibria uninteresting? 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

23. Frank Cowell: Microeconomics Out-of-equilibrium beliefs: a criterion Is an equilibrium at z a > z 0 “reasonable”? Is an equilibrium at z a > z 0 “reasonable”?  requires w() that sets w(z′) <  (  a ) for z 0 < z′ < z a  so firms must be assigning the belief π(z′)>0 Imagine someone observed choosing z′ Imagine someone observed choosing z′  b-type IC through (z′,  (  a )) lies below the IC through (0,  (  b ))  a b-type knows he’s worse off than in the separating equilibrium  a b-type would never go to (z′,  (  a ))  so anyone at z′ out of equilibrium must be an a-type. An intuitive criterion: An intuitive criterion:  π(z′) = 0 for any z′  (z 0, z a ) So only separating equilibrium worth considering is where So only separating equilibrium worth considering is where  a-types are at (z 0,  (  a ))  b-types are at (0,  (  b )).

24. Frank Cowell: Microeconomics 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

25. Frank Cowell: Microeconomics Pooling There may be equilibria where the educational signal does not work 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  …  …and relationship between marginal productivity and  All workers present themselves with the same credentials 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  in the population Firms’ beliefs are derived from the distribution of  in the population  this distribution is common knowledge So wage offered is expected marginal productivity So wage offered is expected marginal productivity  E  (  ):=[1   ]  (  a ) +  (  b ) Being paid this wage might be in interests of all workers… Being paid this wage might be in interests of all workers… Example

26. Frank Cowell: Microeconomics 0 z y v(,b)v(,b) z0z0 v(,a)v(,a) z1z1 (a)(a) (b)(b) E()E() 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, )  both pass through (0, E  (  ))  the type-b IC must be higher than with signalling  but, in this case, so is the type-a IC z0z0  should school be banned?

27. Frank Cowell: Microeconomics   critical z-value for b-type to accept pooling payoff 0 z y v(,b)v(,b) z2z2 (a)(a) (b)(b) E()E() Pooling: limits on z?   critical IC for a b-type   (  ) = [1  ]  (  a ) +  (  b )  E  (  ) = [1  ]  (  a ) +  (  b )   expected marginal productivity  [1  ]  (  a ) +  (  b )  C(z 2,  b ) =  (  b )   b-type payoff with 0 education   viable z-values in pooling eqm

28. Frank Cowell: Microeconomics Pooling equilibrium: example 1 0 z y v(,b)v(,b)v(,a)v(,a) w() z*z* (a)(a) (b)(b) E()E()   expected marginal productivity   viable z-values in pooling eqm   wage schedule   utility maximisation   equilibrium education

29. Frank Cowell: Microeconomics Pooling equilibrium: example 2 0 z y v(,b)v(,b)v(,a)v(,a) w() z*z* (a)(a) (b)(b)   expected marginal productivity   viable z-values in pooling eqm   wage schedule   utility maximisation   equilibrium education E()E()   but is pooling consistent with out-of-equilibrium behaviour?

30. Frank Cowell: Microeconomics 0 z y v(,b)v(,b) z0z0 v(,a)v(,a) (a)(a) (b)(b) E()E() z'z'z*z* Intuitive criterion again   a pooling equilibrium   a critical z-value z'   C(z *,  b ) =  (  a )  C(z′,  b )  E  (  )  C(z *,  b ) =  (  a )  C(z′,  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  (z 0 ) = 0   a-type gets higher utility at z 0   would move from z* to z 0   so pooling eqm inconsistent with the intuitive criterion

31. Frank Cowell: Microeconomics Overview... Costly signals: model Costly signals: equilibrium Costless signals Signalling An argument by example

32. Frank Cowell: Microeconomics Costless signals: an example Present the issue with a simplified example Present the issue with a simplified example  general treatments can be difficult N risk-neutral agents share in a project with output N risk-neutral agents share in a project with output  q =  [z 1 ×z 2 ×z 3 ×...] where 0 < α < 1  z h = 0 or 1 is participation indicator of agent h Agent h has cost of participation c h (unknown to others) Agent h has cost of participation c h (unknown to others)  c h  [0,1]  it is common knowledge that prob(c h ≤ c) = c Output is a public good, so net payoff to each agent h is Output is a public good, so net payoff to each agent h is q  chq  chq  chq  ch Consider this as a simultaneous-move game Consider this as a simultaneous-move game  what is the NE?  improve on NE by making announcements before the game starts?

33. Frank Cowell: Microeconomics Example: NE without signals Central problem: each h risks incurring cost c h while getting consumption 0 Central problem: each h risks incurring cost c h while getting consumption 0 If π is the probability that any other agent participates, payoff to h is If π is the probability that any other agent participates, payoff to h is   −c h with probability [  ] N−1  −c h otherwise Expected payoff to h is  [  ] N−1 − c h Expected payoff to h is  [  ] N−1 − c h Probability that expected payoff is positive is  [  ] N−1 Probability that expected payoff is positive is  [  ] N−1  but this is the probability that agent h actually participates  therefore  =  [  ] N−1  this can only be satisfied if  = 0 So the NE is z h = 0 for all h, as long as α < 1 So the NE is z h = 0 for all h, as long as α < 1

34. Frank Cowell: Microeconomics Example: introduce signals Introduce a preliminary stage to the game Introduce a preliminary stage to the game Each agent has the opportunity to signal his intention: 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 Then there is an equilibrium in which the following occurs  each h announces [YES] if and only if c h < α  h selects z h = 1 iff all agents have announced [YES] In this equilibrium: In this equilibrium:  agents don’t risk wasted effort  if there are genuine high-cost c h agents present that inhibit the project…  …this will be announced at the signalling stage

35. Frank Cowell: Microeconomics Signalling: summary Both costly and costless signals are important Both costly and costless signals are important Costly signals: 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: Costless signals:  a role to play in before the game starts