#### Transcript Extensive Form - London School of Economics

Prerequisites Almost essential Risk Frank Cowell: Microeconomics February 2007 Signalling MICROECONOMICS Principles and Analysis Frank Cowell Introduction Frank Cowell: Microeconomics Jump to “Moral Hazard” Jump to “Adverse selection” A key aspect of hidden information Information relates to personal characteristics But a fundamental difference from screening hidden information about actions is dealt with under “moral hazard” 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) Signalling Frank Cowell: Microeconomics Agent with the information makes first move: Types of signal subtly different from other “screening” problems move involves making a 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”… Talent Frank Cowell: Microeconomics Suppose individuals differ in terms of hidden talent τ Talent is valuable in the market If a signal is not possible but possessor of τ cannot convince buyers in the market without providing a signal that he has it may be no market equilibrium If a signal is possible will there be equilibrium? …more than one equilibrium? Overview... Signalling Frank Cowell: Microeconomics Costly signals: model An educational analogy Costly signals: equilibrium Costless signals Costly signals Frank Cowell: Microeconomics Suppose that a “signal” costs something Consider a simple model of the labour market Suppose productivity depends on ability the able – ta the bog-standard – tb ta > tb Single type of job Ability is not observable Two types of workers: physical investment… forgone income… 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? Signals: educational “investment” Frank Cowell: Microeconomics Consider the decision about whether acquire education Suppose talent on the job identical to talent at achieving educational credentials Education does not enhance productive ability assumed to be common knowledge may be worth “investing” in the acquisition of credentials. 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 Signalling by workers “Nature” determines worker’s type Workers decide on education 0 1-p [LOW] [HIGH] h [NOT INVEST] h Firms make wage offers Workers decide whether to accept [NOT [INVEST] INVEST] [INVEST] f1 [low] [high] [low] [high] f2 [low] [high] [low] [high] [low] [high] [low] [high] investment involves time and money simultaneous offers: Bertrand competition h [accept 1] Frank Cowell: Microeconomics p … … … Examine stages 1-3 more closely A model of costly signals Frank Cowell: Microeconomics Previous sketch of problem is simplified 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 workers only make binary decisions (whether or not to invest) firms only make binary decisions (high or low wage) C(0, t) = 0 Czz(z, t) > 0 Cz(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 Costly signals Frank Cowell: Microeconomics (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) 0 z z0 C(z, t) = (1/t) z2 y 18 16 low t 14 12 Payoffs to individuals 10 8 6 Frank Cowell: Microeconomics high t 4 2 z 0 0 1.5 2 2.5 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: Example 1 Talent does not enter the worker's utility function directly 0.5 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 3 3.5 Rational behaviour Frank Cowell: Microeconomics Workers: Wage is conditioned on “signal” that they provide through acquisition of educational credentials Type-τ worker chooses z to maximise assume income y is determined by wage 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? Overview... Signalling Frank Cowell: Microeconomics Costly signals: model Costly signals discriminate among agents Costly signals: equilibrium Costless signals •Separating equilibrium •Out-of-equilibrium behaviour •Pooling equilibrium Separating equilibrium (1) Frank Cowell: Microeconomics Start with a separating Perfect Bayesian Equilibrium Both type-a and type-b agents are maximising Therefore, for the talented a-types we have f(ta) - C(za, tb) ≤ f(tb) - C(zb, tb) Rearranging this 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: so neither wants to switch to using the other's signal 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 Separating equilibrium (2) Frank Cowell: Microeconomics If there are just two types, at the optimum zb = 0 So, conditions for separating equilibrium become remember that C(za, ta) ≤ f(ta) - f(tb) C(za, tb) ≥ f(ta) - f(tb) C(0,t)=0 Let z0, z1 be the critical z-values that satisfy these conditions with equality everyone knows there are only two productivity types education does not enhance productivity so no gain to b-types in buying education 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… Bounds to education Frank Cowell: Microeconomics 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) f(tb) 0 Separating eqm: Two examples z0 z1 z Separating equilibrium: example 1 Frank Cowell: Microeconomics “bounding” ICs for each type possible equilibrium z-values wage schedule y max type-b’s utility v(•,ta) max type-a’s utility both curves pass through (0, f(tb)) • f(ta) determines z0, z1 as before w(•) low talent acquires zero education v(•,tb) f(tb) high talent acquires education close to z0 • 0 za z Separating equilibrium: example 2 Frank Cowell: Microeconomics possible equilibrium z-values a different wage schedule y max type-b’s utility max type-a’s utility v(•,ta) • f(ta) just as before low talent acquires zero education (just as before) high talent acquires education close to z1 w(•) v(•,tb) f(tb) • 0 za z Overview... Signalling Frank Cowell: Microeconomics Costly signals: model More on beliefs… Costly signals: equilibrium Costless signals •Separating equilibrium •Out-of-equilibrium behaviour •Pooling equilibrium Out-of-equilibrium-beliefs: problem Frank Cowell: Microeconomics For a given equilibrium can redraw w(⋅)-schedule Shape of the w(⋅)-schedule at other values of z? resulting attainable set for the workers must induce them to choose (za, f(ta)) and (0, f(tb)) 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 Perfect Bayesian Equilibria Frank Cowell: Microeconomics Requirements for PBE do not help us to select among the separating equilibria Education level z0 is the minimum-cost signal for a-types try common sense? 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 Paretodominated outcomes Need a proper argument, based on the reasonableness of such an equilibrium Out-of-equilibrium beliefs: a criterion Frank Cowell: Microeconomics Is an equilibrium at za > z0 “reasonable”? 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: requires w(•) that sets w(z′) < f(ta) for z0 < z′ < za so firms must be assigning the belief π(z′)>0 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)). Overview... Signalling Frank Cowell: Microeconomics Costly signals: model Agents appear to be al the same Costly signals: equilibrium Costless signals •Separating equilibrium •Out-of-equilibrium behaviour •Pooling equilibrium Pooling Frank Cowell: Microeconomics 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 Example Ef(t):=[1 - p]f(ta) + pf(tb) Being paid this wage might be in interests of all workers… No signals: an example Frank Cowell: Microeconomics possible z-values with signalling outcome under signalling y outcome without signalling v(•,tb) highest a-type IC under signalling v(•,ta) f(ta) Ef(t) 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(tb) 0 zz00 z z1 should school be banned? Pooling: limits on z? Frank Cowell: Microeconomics 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 0 z2 Pooling equilibrium: example 1 Frank Cowell: Microeconomics expected marginal productivity viable z-values in pooling eqm v(•,tb) y v(•,ta) wage schedule utility maximisation equilibrium education f(ta) w(•) Ef(t) f(tb) 0 z* z Pooling equilibrium: example 2 Frank Cowell: Microeconomics expected marginal productivity viable z-values in pooling eqm v(•,tb) y v(•,ta) wage schedule utility maximisation equilibrium education f(ta) but is pooling consistent with out-of-equilibrium behaviour? w(•) Ef(t) f(tb) 0 z* z Intuitive criterion again Frank Cowell: Microeconomics a pooling equilibrium a critical z-value z' y wage offer for an a-type at z0 > z' max b-type utility at z0 v(•,ta) v(•,tb) f(ta) Ef(t) Ef(t) - C(z*, tb) = f(ta) - C(z′,tb) 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 z with the intuitive criterion f(tb) 0 max a-type utility at z0 z* z' z0 Overview... Signalling Frank Cowell: Microeconomics Costly signals: model An argument by example Costly signals: equilibrium Costless signals Costless signals: an example Frank Cowell: Microeconomics Present the issue with a simplified example N risk-neutral agents share in a project with output 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 = 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) general treatments can be difficult q - ch Consider this as a simultaneous-move game what is the NE? improve on NE by making announcements before the game starts? Example: NE without signals Frank Cowell: Microeconomics 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 h −c 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 Example: introduce signals Frank Cowell: Microeconomics Introduce a preliminary stage to the game Each agent has the opportunity to signal his intention: Then there is an equilibrium in which the following occurs each agent announces [YES] or [NO] to the others each agent then decides whether or not to participate 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 Signalling: summary Frank Cowell: Microeconomics 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