Signalling

1. Prerequisites Almost essential Risk Signalling MICROECONOMICS Principles and Analysis Frank Cowell February 2007

2. 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) Jump to “Moral Hazard” Jump to “Adverse selection”

3. 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”…

4. 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?

5. Overview... Signalling Costly signals: model An educational analogy Costly signals: equilibrium Costless signals

6. 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 bog-standard – 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?

7. 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

8. f1 [low] [high] [low] [high] f2 [high] [high] [high] [high] [low] [low] [low] [low] Signalling by workers 0 • “Nature” determines worker’s type • Workers decide on education p 1-p • Firms make wage offers [LOW] [HIGH] h h • Workers decide whether to accept [NOT INVEST] [NOT INVEST] [INVEST] [INVEST] • investment involves time and money • simultaneous offers: Bertrand competition Examine stages 1-3 more closely h … … … [reject] [accept 1] [accept 2]

9. 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

10. Costly signals • (education, cost)-space • Cost function for an a type C • Cost function for a b type • Costs of investment z0 • MC of investment z0 C(•,tb) C(z0,ta) C(•,ta) C(z0,tb) z 0 z0

11. C(z, t) = (1/t) z2 y 18 16 low t 14 12 10 8 6 high t 4 2 z 0 0 0.5 1 1.5 2 2.5 3 3.5 Payoffs to individuals • 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; 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 problem Example

12. Rational behaviour • Workers: • 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?

13. Overview... Signalling Costly signals: model Costly signals discriminate among agents • Separating equilibrium • Out-of-equilibrium behaviour • Pooling equilibrium Costly signals: equilibrium Costless signals

14. 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

15. 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

16. Bounds to education • IC for a b type • IC for an a type y • critical value for an a type • critical value for a b type • possible equilibrium z-values v(•,ta) • both curves pass through (0, f(tb)) • f(ta) = f(tb)  C(z1, ta) f(ta) • f(ta) = f(tb)  C(z0, tb) v(•,tb) Separating eqm: Two examples f(tb) z 0 z0 z1

17. Separating equilibrium: example 1 • “bounding” ICs for each type • possible equilibrium z-values y • wage schedule • max type-b’s utility • max type-a’s utility v(•,ta) • both curves pass through (0, f(tb)) • determines z0, z1 as before • f(ta) w(•) • low talent acquires zero education v(•,tb) • high talent acquires education close to z0 • f(tb) z 0 za

18. Separating equilibrium: example 2 • possible equilibrium z-values • a different wage schedule y • max type-b’s utility • max type-a’s utility v(•,ta) • just as before • low talent acquires zero education (just as before) • high talent acquires education close to z1 • f(ta) w(•) v(•,tb) • f(tb) z 0 za

19. Overview... Signalling Costly signals: model More on beliefs… • Separating equilibrium • Out-of-equilibrium behaviour • Pooling equilibrium Costly signals: equilibrium Costless signals

20. Out-of-equilibrium-beliefs: problem • For 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 equilibrium • PBE leaves open what out-of-equilibrium beliefs may be

21. 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

22. 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)).

23. Overview... Signalling Costly signals: model Agents appear to be al the same • Separating equilibrium • Out-of-equilibrium behaviour • Pooling equilibrium Costly signals: equilibrium Costless signals

24. 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 • Ef(t):=[1 p]f(ta) + pf(tb) • Being paid this wage might be in interests of all workers… Example

25. No signals: an example • possible z-values with signalling y • outcome under signalling • outcome without signalling v(•,tb) • highest a-type IC under signalling v(•,ta) • both pass through (0, Ef(t)) • the type-b IC must be higher than with signalling • but, in this case, so is the type-a IC f(ta) • Ef(t) f(tb) • should school be banned? z z0 0 z0 z1

26. Pooling: limits on z? • b-type payoff with 0 education • critical IC for a b-type y • expected marginal productivity • critical z-value for b-type to accept pooling payoff v(•,tb) • viable z-values in pooling eqm • Ef(t) = [1p]f(ta) + pf(tb) f(ta) • [1p]f(ta) + pf(tb)  C(z2, tb) = f(tb) Ef(t) f(tb) z z2 0

27. Pooling equilibrium: example 1 • expected marginal productivity • viable z-values in pooling eqm • wage schedule v(•,tb) v(•,ta) y • utility maximisation • equilibrium education w(•) f(ta) Ef(t) f(tb) z 0 z*

28. Pooling equilibrium: example 2 • expected marginal productivity • viable z-values in pooling eqm • wage schedule v(•,tb) v(•,ta) y • utility maximisation • equilibrium education • but is pooling consistent with out-of-equilibrium behaviour? w(•) f(ta) Ef(t) f(tb) z 0 z*

29. Intuitive criterion again • a pooling equilibrium • a critical z-value z' y • wage offer for an a-type at z0 > z' • max b-type utility at z0 • max a-type utility at z0 v(•,ta) • Ef(t) C(z*, tb) = f(ta) C(z′,tb) v(•,tb) • b-type would not choose z0 • under intuitive criterion p(z0) = 0 f(ta) Ef(t) • a-type gets higher utility at z0 • would move from z* to z0 • so pooling eqm inconsistent with the intuitive criterion f(tb) z z* z' z0 0

30. Overview... Signalling Costly signals: model An argument by example Costly signals: equilibrium Costless signals

31. 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?

32. 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

33. 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

34. 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