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

Frank Cowell: Signalling 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

Frank Cowell: Signalling 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

Frank Cowell: Signalling 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

Frank Cowell: Signalling Overview Costly signals: model Costly signals: equilibrium Costless signals Signalling An educational analogy July

Frank Cowell: Signalling 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 –  a the basic –  b  a >  b  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? July

Frank Cowell: Signalling 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

Frank Cowell: Signalling 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 July

Frank Cowell: Signalling 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   Cost of acquiring education level z is C(z,  ) ≥ 0 C(0,  ) = 0C z (z,  ) > 0 C zz (z,  ) > 0C z  (z,  ) < 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

Frank Cowell: Signalling 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 July

Frank Cowell: Signalling 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;  ) := 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: 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 Example y z high  low  C(z,  ) = (1/  ) z 2 July

Frank Cowell: Signalling 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,  ) 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

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

Frank Cowell: Signalling 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  (  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:  (  a )  C(z a,  b ) ≤  (  b )  C(z b,  b )  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 July

Frank Cowell: Signalling Separating equilibrium (2)  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 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 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 remember that C(0,  )=0 July

Frank Cowell: Signalling 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,  (  b ))  possible equilibrium z -values   (  a ) =  (  b )  C(z 1,  a )   (  a ) =  (  b )  C(z 0,  b ) Separating eqm: Two examples July

Frank Cowell: Signalling 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,  (  b ))  determines z 0, z 1 as before  low talent acquires zero education z y  high talent acquires education close to z 0 July

Frank Cowell: Signalling 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 July

Frank Cowell: Signalling Overview Costly signals: model Costly signals: equilibrium Costless signals Signalling More on beliefs Separating equilibrium Out-of-equilibrium behaviour Pooling equilibrium July

Frank Cowell: Signalling Out-of-equilibrium-beliefs: problem  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? 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 July

Frank Cowell: Signalling Perfect Bayesian 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 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? important cases of strategic interaction that produce Pareto-dominated outcomes need a proper argument, based on the reasonableness of such an equilibrium July

Frank Cowell: Signalling Out-of-equilibrium beliefs: a criterion  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′ 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: π(z′) = 0 for any z′  (z 0, z a )  So only separating equilibrium worth considering is where a-types are at (z 0,  (  a )) b-types are at (0,  (  b )) July

Frank Cowell: Signalling 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 July

Frank Cowell: Signalling 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  and relationship between marginal productivity and   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 this distribution is common knowledge  So wage offered is expected marginal productivity E  (  ):=[1   ]  (  a ) +  (  b )  Being paid this wage might be in interests of all workers Example July

Frank Cowell: Signalling 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, 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? July

Frank Cowell: Signalling  critical z 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  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 July

Frank Cowell: Signalling 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 July

Frank Cowell: Signalling 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? July

Frank Cowell: Signalling 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'  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 intuitive criterion July

Frank Cowell: Signalling Overview Costly signals: model Costly signals: equilibrium Costless signals Signalling An argument by example July

Frank Cowell: Signalling 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 =  [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) 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 q  c h  Consider this as a simultaneous-move game what is the NE? improve on NE by making announcements before the game starts? July

Frank Cowell: Signalling Example: NE without signals  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  −c h with probability [  ] N−1 −c h otherwise  Expected payoff to h is  [  ] N−1 − c h  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 July

Frank Cowell: Signalling 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 c h < α h selects z h = 1 iff all agents have announced [YES]  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 July

Frank Cowell: Signalling 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