Adverse Selection, Signaling, and Screening in Markets

BGPE Intensive Course: Contracts and AsymmetricInformation

Adverse Selection, Signaling, and Screening inMarkets

Anke Kessler

Anke Kessler – p. 1/27

Market Failure

Stylized Facts:

• used cars, even if they are like new, sell far below their dealershipprice

• laid-off workers experience longer spells of unemployment thanworkers for different reasons without a job (e.g. military)

• private health care for the elderly is essentially unavailable

• young drivers pay very expensive insurance premiums

what do these empirical regularities have in common?

these markets are characterized by informational asymmetries andsuffer from adverse selection


Productivity Uncertainty in the Labor Market

• 2 identical firms F , 1 representative worker W

• worker’s productivity is θ ∈ [θ, θ], private information of W , c.d.f F (θ)is continuous and has full support

• utilitiesuW = wq + r(θ)(1 − q), uF = (θ − w)q

where q = 1 if W works and q = 0 otherwisew ∈ IR = wage paid by firmr(θ) = W ’s reservation utility, r′ > 0

• timing:• t = 0: W observes θ

• t = 1: F offer fixed-wage employment contracts• t = 2: W accept/rejects → payoffs


Equilibrium Analysis

Equilibrium under Full Information

• both worker and firms know θ → firms can offer productivitydependent wages w(θ)

• competition between firms gives w∗(θ) = θ in equilibrium

• W accepts employment if w∗(θ) = θ ≥ r(θ)

• equilibrium is efficient

Equilibrium under Asymmetric Information

• only W knows θ, F ’s only know F (θ)

• assume r(θ) ≤ θ and r(θ) < θ, ∀θ > θ

⇒ efficient employment has q(θ) = 1 ∀θ


Equilibrium under Asymmetric Information

• firms do not know θ ⇒ offer (same) fixed wage contract w inequilibrium

• utility of worker of type θ who accepts wage offer of w is uW = w

• comparing this with outside utility r(θ) gives supply

Θ(w) = {θ| r(θ) ≤ w}

note: average productivity E{θ|θ ∈ Θ(w)} depends on price

• expected profit of firm in the market is E{θ|θ ∈ Θ(w)} − w

• competition gives zero profits; necessary equilibrium condition is

w∗ = E{θ|θ ∈ Θ(w∗)} = E{θ|r(θ) ≤ w∗}


Market Equilibrium

• graphic illustration:

45◦

θ

E(θ)

r(θ) w∗

r(θ)w

E{θ|r(θ) ≤ w∗}


Properties of the Equilibrium

• equilibrium exists

• equilibrium is generically unique: if there are multiple solutions toequation w∗ = E{θ|θ ∈ Θ(w∗)} = E{θ|r(θ) ≤ w∗}, equilibrium ishighest w satisfying condition

• welfare properties:

• if r(θ) ≤ E(θ), equilibrium is efficient

• if r(θ) > E(θ), equilibrium is inefficient• if r(θ) = θ, can have complete market breakdown

• intuition: F cannot break even at wage w = E(θ) < r(θ)w falls → even more high-productivity workers drop out of market →wage must drop further(alternatively : market participation of individual worker introducesexternality)


Government Intervention

Can free-market outcome be improved upon?

• assume: social planner/government does not know workers’productivity θ either (is subject to same informational constraints asfirms)

→ can only devise policies based on whether people choose to work ornot (from IC’s)

• let we (resp. wu) be transfer to worker if q = 1 (resp. q = 0)

wu + r(θ) = we (IC)

Claim. There is no (balanced) policy {we, wu} that yields to a Paretoimprovement over the market outcome where workers with θ ≤ θ∗ workand workers θ > θ∗ do not.


Government Intervention

Proof.• equilibrium is Pareto efficient if r(θ) ≤ E(θ)

• if r(θ) > E(θ), let θ be cut-off type worker given {we, wu}

• budget balance requires

weF (θ) + wu(1 − F (θ)) =

Z θ

θ

θf(θ)dθ = F (θ)E(θ|θ ≤ θ)

• substituting for wu + r(θ) = we gives

wu(θ) = F (θ)[E(θ|θ ≤ θ) − r(θ)]

we(θ) = F (θ)[E(θ|θ ≤ θ) − r(θ)] + r(θ)

• note: θ = θ∗ gives market outcome wu = 0 and we = r(θ∗)

• θ < θ∗ decreases social surplus

• suppose θ∗ < θ < θ. Since w > w∗ ⇒ E(θ|r(θ) ≤ w) < w and w∗ = r(θ∗) withr′ > 0, we get E(θ|r(θ) ≤ r(θ)) = E(θ|θ ≤ θ) < r(θ) and, hence, wu < 0

→ unemployed are worse off

• θ = θ implies we = E(θ) → highest type workers are worse off since r(θ) > E(θ). �


Other Markets with Adverse Selection

• formal model translates into goods markets and insurance markets:labor consumption good (cars) insurance

workers → sellers insureefirms → buyers insurer

θ → buyer’s value -(exp. payments to insuree)r(θ) → seller’s value -(inverse CE of risk)

w → price -(insurance premium)

• other markets where adverse selection relevant• credit market (θ = default risk of debtor)• dating and marriage market (θ = attractiveness of partner)• stock market and corporate equity market (IPO’s) (θ = firm value)


Conclusion

adverse selection can lead to total market failure – if trade occurs,it will be less than efficient

• in markets with adverse selection (asymmetric information)

• prices are correlated with quality

• prices serve dual role of info transmission and market clearing

• insitutional/market responses against market failure caused byadverse selection• signaling and screening devices, e.g. warrantees• reputation (brand names and chains)• experts, inspections, standards, licensing• mandatory insurance (health, automobile)• liability laws


Using a Signal

asymmetric information causes market failure → participants haveincentives to develop ways to reduce informational asymmetries

• signaling: informed market participants move first to convey info

• screening: uninformed market participants move first to elicit info

Signaling

• some market participants may be worse off as a result of theirprivately held information ( sellers in lemons market, consumers ininsurance market) → would want to reveal this information to others

• problem: information revealed must be credible → use of signals

• examples: warranties, lineups, peacock tail

• but: for the signal to work (be credible), it must be costly to fake


Education as a Signal in Labor Markets (Spence)


• worker W with ability (=produtivity) θ ∈ {θL, θH}, θL < θH , privateinformation of W with p = Prob{θ = θH}

• set r(θH) = r(θL) = 0 for simplicity

• worker can invest in education e ∈ IR+0

• marginal cost of education of θi-type is ci with 0 < cH < cL

• education does not improve productivity θ

• utilities:

uF = θi − w and uW = w − cie i = H, L• timing:

t = 1: W learns θi, chooses e

t = 2: F ’s observe e, form beliefs µ(e) = Prop{θ = θH |e}t = 3: F ’s offers (same) wage w


General Structure of Signaling (and Cheap Talk) Game

• dynamic game of incomplete information with sender S and receiver R

• timing

t = 0 nature draws type θi ∈ Θ = {θ1, θ2, . . . , θn} for sender; with

pi = p(θi) = Prob{θ = θi} > 0, ∀θi ∈ Θ

t = 1 sender observes own type θi, chooses message

mj ∈ M = {m1, m2, . . . , mN}

t = 2 receiver observes message mj (but not θi) andchooses action x ∈ X

t = 3 payoffs uS(x, mj , θi) and uR(x, mj , θi) realized

• signaling game: uS depends on mj

• cheap talk game: uS (and uR) is independent of mj


Perfect Bayesian Nash Equilibrium

Definition A Perfect Bayesian Equilibrium (PBE) is a pair of strategiesm∗(θi) and x∗(mj) and a belief µ∗(θi|mj) such that

a) m∗(θi) maximizes S’s utility given R’s strategy:

m∗(θi) = arg maxmj∈M uS(x, mj , θi) ∀θi ∈ Θ

b) x∗(mj) maximizes R’s utility given beliefs about S’s type:

x∗(mj) = arg maxx∈X

∑

θi

µ(θi|mj)uR(x, mj , θi),

c) R forms consistent beliefs that are calculated by Bayes Rulewhenever possible:

∑

θi∈Θ

µ(θi|mj) = 1, ∀mj µ(θi|mj) =p(θi)∑

θi∈Θ(mj)p(θi)

where Θ(mj) ≡ {θi ∈ Θ|m∗(θi) = mj}.


Benchmark Cases


• both worker and firms observe ability θi, i = H, L

• firms offer wage wi = θi independent of e

→ worker chooses eH = eL = 0 ⇒ efficient

Imperfect but Symmetric Information

• neither worker nor firms observe θi, i = H, L

→ wage w can no longer depend on θ

• firms offer w(e = 1) = w(e = 0) = pθH + (1 − p)θL independent of e

→ worker chooses eH = eL = 0 ⇒ efficient


Imperfect and Asymmetric Information

• only worker not firms observe θi, i = H, L


Separating Equilibrium

• two types of workers choose different education levels eH 6= eL

→ firms have beliefs µ(eH) = θH and µ(eL) = θL

→ firms offer wages w(eH) = θH and w(eL) = θL

• eH 6= eL optimal for W requires (from (IC)’s) eL = 0 and eH > 0where

cHeH < θH − θL ≤ cLeH (⋆)

• off equilibrium beliefs? ... let µ(e) = 0 for e < eH , µ(e) = 1 for e ≥ eH


Education as a Signal

• graphic illustration of a separating equilibrium:

{(w, e)|u(w, e; θL) = u(wH , eH ; θL)}

{(w, e)|u(w, e; θH ) = u(wH , eH ; θH )}

θL

0

θH

eH

w

e

• note: W ’s preferences satisfy single crossing property:

dw

de|u ↑ in θ or ue(w, e; θL) − ue(w, e, ; θH) = cL − cH > 0 (SC)


Other Types of Equilibria

Pooling Equilibrium

• both types of workers choose same education level eH = eL = e

→ firms have beliefs µ(e) = pθH + (1 − p)θL

→ firms offer wage w(e) = θH + (1 − p)θL

• off equilibrium beliefs? ... let µ(e) = 0 for e < e, µ(e) = p for e ≥ e

• note: both e = 0 and e > 0 may be possible

Semi-Separating Equilibrium

• some education levels are chosen by both types, and some by onlyone type

• no type chooses more than two levels of e with positive probability

• three types of semi-separating equilibria, depending on who makestwo different choices with positive probability


Properties of Signaling in Markets

• equilibrium with eH = eL = 0 always exists

• wasteful signaling in separating equilibria where eH > 0

• wasteful education also possible in some pooling equilibria whereeH = eL > 0

• multiple (types) of equilibria cannot be unambiguously ranked by thePareto criterion

• out-of-equilibrium beliefs µ(e) are critical but not always plausible →equilibrium refinements

Intuitive Criterion (Cho-Kreps): if an out-of-equilibrium deviation is dominated for some typeθi, but not for all types, then out-of-equiloibrium beliefs should put zero probability of that type

action “dominated” = whatever the receiver’s action to the deviation, type θi is made worseoff by the deviation

→ selects unique least-cost separating equilibrium

this equilibrium may be Pareto dominated, however (intuition = informational externality)


Conclusion

signaling can least to wasteful resource allocation and the marketoutcome may thus be inefficient

• in markets with signaling• privately informed individuals use signal to reveal their

information• signal only works (is credible) if sending the same signal is too

costly for other individuals

• other markets where wasteful signaling is relevant• consumer products (signal = warranty, advertisements, price)

• corporate equity and start-ups (signal = equity/own money invested)

• legal disputes (signal = pre-trial settlement demands)

• bargaining (signal = rejection of offer/delay)

• live entertainment and restaurants (signal = lineups)

• marriage and dating (signal = fancy car)

• poker (signal = stakes)


Competitive Screening

Screening = contractual arrangements originating from uniformed sideof market to elicit information from informed market participants


• worker W with ability (=produtivity) θ ∈ {θL, θH}, θL < θH , privateinformation of W with p = Prob{θ = θH}, r(θi) = 0

• firm can set task difficulty level t ∈ IR+0

• utilities: uF = θi − w and uW = w − cit i = H, L

• task difficulty does not influence productivity and costs workerseffort ci with 0 < cH < cL

• timing:t = 1: W learns θi

t = 2: F ’s offer menu of contracts {(w, t)}t = 3: W picks firm/contract


Equilibrium Analysis

• note: W ’s preferences satisfy single crossing property:

dw

dt|u ↑ in θ or ut(w, t; θL) − ut(w, t, ; θH) = cL − cH > 0 (SC)


• both worker and firms observe ability θi, i = H, L

• firms offer wage/task contracts (wi, ti) = (θi, 0)

→ worker accepts employment ⇒ SPE is efficient

Imperfect but Symmetric Information

• neither worker nor firms observe θi, i = H, L


• firms offer wage/task contract (w, t) = (pθH + (1 − p)θL, 0)

→ worker accepts ⇒ efficient


Imperfect and Asymmetric Information

• only worker not firms observe θi, i = H, L


find equilibrium in steps

• let (wH , tH) and (wL, tL) be the lowest-wage contracts accepted byworkers of type i in equilibrium

• Step 1: in any SPE, firms earn zero profits

• Step 2: there is no SPE in which the high-type worker accepts awage wH < θH with positive probability

→ all low types must earn wL = θL and the equilibrium must beseparating

• in a separating SPE, must have tL = 0 and θH − cLtH = θL, i.e., theonly candidate for equilibrium is the least-cost separating one


Screening with Task Assignments

• graphic illustration of the separating equilibrium:

type θL

type θH(wH , tH )

(wL, tL)

E[θ]

θL

0

θH

w

t

• note: no deviation profitable; in particular, any contract attractingboth types of workers lies above break even line


Screening with Task Assignments

• graphic illustration of the separating equilibrium:

type θL

type θH

(wH , tH )

(wL, tL)

(w, t)

E[θ]

θL

0

θH

w

t

• note: deviation profitable; a contract (w, t) attracting both types ofworkers lies below break even line

• if least-cost separating equilibrium is Pareto dominated by pooling,then no equilibrium exists


Properties of Market Screening

• equilibrium (in pure strategies) may not exist

• if equilibrium exist, is unique and identical to least-cost separatingequilibrium

• equilibrium separation is necessary; otherwise, rival firms can do“cream skimming” and attract only high-types away form the firm

→ wasteful screening with tH > 0 occurs in equilibrium → outcome isPareto inefficient

• other screening devices in labor markets• probation periods• seniority wages• performance based compensation


Conclusion

sceening can least to wasteful resource allocation and the marketoutcome may thus be inefficient

• in markets with screening• uninformed individuals use screening devices to make informed

individuals reveal their information by choice of (preferred)contract from menu

• screening device only works (separates types) if accepting samecontract is undesirable for other individuals

• other markets where wasteful screening is relevant• consumer products (screening device = warranty, price)

• credit (screening device = collateral, incomes)

• marriage and dating (screening device = household chores)

• poker (screening device = stakes)


Adverse Selection, Signaling, and Screening in Markets

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