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Microeconomics - 3.1 Private Information

Adverse Selection Signaling

Microeconomics

3. Information Economics

Alex Gershkov

http://www.econ2.uni-bonn.de/gershkov/gershkov.htm

19. Dezember 2007

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Information Economics

We will apply the tools developed in game theory to analyse theeffect of informational asymmetries in the models of:

1.a) adverse selection

1.b) signaling

1.c) screening.

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1.a Adverse Selection

DefinitionAdverse Selection arises when an informed individual’s tradingdecision depends on privately held information in a way thatadversely affects other uninformed market participants.

When adverse selection is present, uninformed traders will be waryof any informed trader who wishes to trade with them, and theirwillingness to pay for the product offered will be low.

Remark: This excludes first-best outcomes.

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Consider the following motivating example

◮ there is a large number of buyers and sellers

◮ each seller has one car

◮ each buyer is willing to buy at most one car

◮ suppose the quality of a car can be indexed by some numberq ∈ [0, 1]

◮ buyers are willing to pay 32q for a car of quality q

◮ sellers are willing to sell a car of quality q for a price of q

◮ assume that q ∼ U[0,1]

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◮ under perfect information, since 32q ≥ q for all q, all cars of

any quality are sold

◮ if q were mutually unknown (but the beliefs are the same forbuyers and seller) any price between 1/2 and 3/4 will lead toefficient allocation

Conclusion: Under symmetric information the outcome is efficient(first-best).

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◮ now q ∼ U[0,1]: since the expected quality of a car for the

whole market is q ≡ E[q] = 12 , only a ‘pooling’ price of

p ≤ 3q2 = 3

4 will be offered by the buyers

◮ but at this price, the top quarter of the whole market will notbe supplied because their known valuation by the sellers ishigher than the pooling price offered by the buyers

◮ therefore buyers need to re-calculate the expected marketquality without the withdrawn top-quality quarter: Theexpected quality of the market is E[q′] = q′ = 3

8 and prices up

to p′ ≤ 3q′

2 = 916 are offered

◮ at this price now only 916 < 3

4 of the whole market will beoffered by the sellers

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◮ therefore buyers again will re-calculate the expected qualityand the market will shrink further

◮ as we can see below, this process is monotonic and will onlystop at q = p = 0

◮ hence the complete market unravels and no car (theprobability of q = 0 is zero) is sold.

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q

p

0 1

1

.5 q, pooled quality

p=q, supplyseparating price

{

q|q ≤ 32 q

}

32 q, pooling price.75

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More formally (without assuming uniform distribution), the supplyof sellers with quality q at price p is

s(q, p) =

{

1 if p ≥ q

0 if p < q.

Buyers know that at price p, only cars with q ≤ p are offered. Anybuyer’s expected utility is given by

3

2E[q|q ≤ p] − p.

Therefore, the buyers’ demand at price p is given by

d(p) =

{

1 if 32 E[q|q ≤ p] ≥ p

0 if 32 E[q|q ≤ p] < p.

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1.b Signaling

DefinitionIn signaling models, agents can take actions to distinguishthemselves from their lower-ability counterparts. The preconditionfor this action to be useful as a signaling device is that itsmarginal cost must depend on the agents’ type.

In screening models, the uninformed firm tries to reduce theagents’ informational rent (after contracting) by offering a menu ofcontracts.

In signaling models, the informed agent tries to convey privateinformation to the uninformed firm (before contracting) throughher signal. Hence the problems of learning and updating of beliefsarise.

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1.b Signaling

Timing

1. Nature selects each agent’s type; this is privately observed

2. agents choose a signalling activity

3. based on the observed signal, firms simultaneously make wageoffers

4. agents select one preferred contract

5. outcomes & payoffs realize.

Since the informed party moves first, there is learning in the game.Hence the appropriate sol’n method is BPNE’qm (and not SGPNE’qm as before).

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1.b Signaling: Spence

In the classic signaling model due to Spence (1974) we consider acompetitive labor market where

◮ workers privately know their productivity r ∈ {rL, rH} withrL < rH

◮ ri depends on the private type θi , i ∈ {L,H}; ass. r ′(·) > 0

◮ 2 firms hire workers on the basis of the expected productivityrL < E[r ] < rH of a worker on the market.

Since competitive firms will compensate workers on the basis ofthe market expectation, it is in the interest of high ability workersto signal their high productivity and sign a better contract.

How can they do that?

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The Spence model:

◮ Education is used as a perfectly useless but observable signalsent by the informed employee to let the employer infer herprivate type.

◮ The basic idea is that it is more costly for a low productivityworker to acquire education than for high productivityworkers. Hence, by acquiring more education, the latter candistinguish themselves.

◮ Notice that the precondition for education to be useful as asignal is that its marginal cost must depend on the agent’stype (as does her productivity).

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Thus the model is closed by assuming that

◮ workers select education e before contracting

◮ workers face a cost of acquiring education c(e, θ) withc(0, θ) = 0, ce(e, θ) > 0, cee(e, θ) > 0, cθ(e, θ) < 0,ceθ(e, θ) < 0

◮ workers are willing to work at any wage whereu(w) = w − c(e, θ) > u = 0

◮ firms maximise profits r − w and hire any workers acceptingw ≤ E[r ]

◮ the firms’ prior beliefs are such thatµi = pr(θ = θi), i ∈ {L,H}

◮ there are 2 firms and one worker (population).

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Negative cross derivativeof the cost functionimplies that it costs thehigh ability type less togain another unit ofeducation.

Hence the low type hasthe steeper indifferencecurves.

e

w

e e + ∆

≻+

u(θH)u(θL)

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More formally, single-crossing arises here because the worker’smarginal rate of substitution between wages and education at anygiven pair (w , e) is (dw/de)u = ce(e, θ), which is decreasing in θsince ceθ(e, θ) < 0.

Let’s start with the first-best set up:

◮ it is obvious that each worker type chooses e = 0 becauseeducation serves no purpose and is costly

◮ hence each worker type gets a wage wi = θi , i ∈ {L,H}.

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1.b Signaling: SpenceNow suppose that productivity is not observable

◮ at t = 1, the worker chooses education level e; let theprobability that type θi chooses education level e be given bypi (e) = pr(e(θi ) = e)

◮ at t = 2 the firms simultaneously made a wage offers. Theoutcome of the game is determined entirely through theemployer’s beliefs of which type of worker she faces; let thesebeliefs be given by the conditional probability µ(θi |e) whichdenotes the firm’s revised beliefs about productivity afterobserving the signal e.

The e’qm wage is then given by the expected productivity

w(e) = µ(θH |e)rH + µ(θL|e)rL ∈ [rL, rH ],∀e ≥ 0.

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The key challenge in solving the problem outlined above is theevolution of the firms’ beliefs given the observed event. There aremany game theoretic solution concepts incorporating this and it isnot clear which one to choose.

Let’s begin with the simplest such solution concept, PerfectBayesian NE’qm, and see if we need more powerful concepts as wego along.

(We will.)

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DefinitionA Perfect Bayesian NE’qm (PBE’qm) in Spence’s signaling modelis a strategy pi (e) for each possible worker type and a set ofconditional beliefs µ(θi |e) for the employer such that

◮ all education levels chosen with positive probability in e’qmmust maximise the worker’s expected payoff: that is, ∀e∗

su.t. pi (e∗) > 0, we have

e∗ ∈ argmaxe

µ(θH |e)wH + µ(θL|e)wL − c(e, θi )

◮ if possible, the posteriors must satisfy Bayes’ rule

µ(θi |e) =pr(e|θi ) pr(θi )

pr(e)=

pi (e)µ(θi )∑

j pj(e)µ(θj )

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◮ otherwise unrestricted posterior beliefs must be specifiedeverywhere

◮ firms pay workers their expected productivity

w(e) = µ(θH |e)rH + µ(θL|e)rL.

Assume that the firms have common beliefs on and off the e’qmpath.

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DefinitionA separating PBE in Spence’s signaling model is an e’qm whereeach type of the worker chooses a different signal in e’qm:ie. eH 6= eL such that µ(θH |eH) = 1 and µ(θL|eL) = 1. In aseparating e’qm in Spence’s signaling model we get wi = ri ,i ∈ {L,H}.

DefinitionA pooling PBE in Spence’s signaling model is an e’qm where eachtype of the worker chooses the same signal in e’qm: ie. eH = eL

such that the offered wage is w = µ(θH |e)rH + µ(θL|e)rL.

(Semi-separating equilibria are defined in the obvious way.)

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Case 1. Separating e’qa

Lemma 1:In any separating PBNE w∗(eH) = rH and w∗(eL) = rL.

Lemma 2:In any separating PBNE eL = 0

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Case 1. Separating e’qa

e

w

w∗L = rL

w∗H = rH

w∗P = E[r ]

u(θH)u(θL)

e∗H e1e

SS

≻+

e∗L = 0

w∗S(e)

The dotted line gives apossible wage scheduleand thus representsoff-e’qm-path beliefs.The problem is thatoff-e’qm we can draw itin whatever way welike—since Bayes’ rulecannot be applied(pr(e) = 0 in e’qm), wecan specify anything wewant.

Thus anything in the redstretch is a separatingPBE’qm.

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More formally, the set of possible e in all separating PBE’qa is

SS ={

(eL, eH)|eL = 0 and eH ∈[

e∗H , e1]}

.

To see this, notice that since the firms pay the workers theirexpected productivity, in all separating equilibria w(eL) = rL andw(eH) = rH .

For the equilibria to be separating it must be the case that

wH − c(eH , θH) ≥ wL, wH − c(eH , θL) ≤ wL

which is the above interval (in e’qm rL = wL & rH = wH ).

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Notice that since

w∗(e) = µ∗(θH |e)wH + (1 − µ∗(θH |e))wL

there is a one-to-one mapping between wages and beliefs (on andoff e’qm path)

µ∗(θH |e) =w∗(e) − rL

rH − rL.

But since many schedules w∗(e) are compatible with, for instance,the outcome on the previous slide, there is no way we can get aunique prediction from PBE’qm.

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But clearly, there are better and worse separating e’qa (for boththe employer and the worker)—the problem is that PBE’qmdoesn’t give us a handle on which to choose!

Notice that

◮ low-ability workers are worse off through the introduction ofeducation: they would get E[r ] if education was forbidden

◮ high-ability workers may profit or loose from the introductionof education (depending on whether E[r ] ≈ rL or E[r ] ≈ rHbefore).

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Case 2. Pooling e’qa

e

w

w∗L = rL

w∗H = rH

w∗P = E[r ]

e∗P

u(θH)u(θL)

e e1

e

≻+

e∗L = 0

w∗P(e)

Again the dotted linegives a possible wageschedule and thusrepresents off-e’qm-pathbeliefs. Again we candraw it in many waysbecause Bayes’ rulecannot be applied(pr(e) = 0 in e’qm) andwe can specify anythingwe want.

Thus anything in the bluestretch is a poolingPBE’qm.

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Similarly, the possible set of e in pooling PBE’qa is

SP = {(eL, eH)|eL = eH ∈ [0, e∗P ]} .

The set’s upper bound derives directly from the low type’s utilitywL ≤ wP − c(e∗P , θL) which, for the e’qm wage wP gives that e∗Pshould satisfy

wL = wP − c(e∗P , θL) = µLrL + µHrH .

Clearly both agents accept anything better, too. Again there arebetter and worse such pooling e’qa from all player’s point of view.The Pareto-dominant one is clearly e∗L = e∗H = 0. But PBE’qm failsto select among them.

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1.b Signaling: SpenceAnything goes: possible PBE’qa

e

w

w∗L = rL

w∗H = rH

w∗P = E[r ]

e∗P

u(θH)u(θL)

e∗H e1

SP

SS

≻+

e∗L = 0

w∗S(e)

w∗P(e)

Belief (wage)schedulessupporting multipleand suboptimalpooling andseparating PBE’qm.

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1.b Signaling: Refinements

Since PBE’qm doesn’t give us any handle on what to believe, we’lllook at stronger criteria:

Cho & Kreps’ (1987) Intuitive Criterion (InC) is based on the ideathat particular deviations from the e’qm path may be in theinterest of some types but not in that of others. Hence we get abetter idea of what to think about the deviator after observing adeviation than in PBE’qm.

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1.b Cho & Kreps (1987)

Definition(Cho & Kreps 1987) Let u∗

i = wi − c(ei , θi ) denote type i ’s e’qm

payoff. Then µ(θj |e) = 0 for an out of e’qm effort e 6= (e∗i ; e∗j ),whenever rH − c(e, θj ) < u∗

j and rH − c(e, θi ) ≥ u∗i , for

i 6= j ∈ {L,H}.

In words: If a deviation is (in e’qm) dominated for one type but notfor another, this deviation should not be attributed to the playerfor whom the deviation is dominated.

Because it is about what to believe after deviations, the InC is arestriction on off-e’qm path beliefs.

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Let’s apply the InC first to pooling e’qa of the Spence model. Weobserve that:

◮ in any pooling e’qm, the 2 types’ indifference curves mustcross as specified by the single crossing property: hence thereis a ‘wedge’ separating high and low types

◮ but then, the high type θH can always find a profitabledeviation (in the wedge) by choosing ed

◮ at ed , the firm offers a wage of w(ed) = rH because, in e’qm,the deviation can only be profitable for a high type (while it isdominated for the low type θL).

This argument eliminates all pooling e’qa and cannot be madeusing PBE’qm!

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The InCdestroys all

poolinge’qa!

Becausewd = rHattractsonly thegood guys.

e

w

ed

wd

wP = E[r ]

eP

u(θL)

u(θL)

u(θH)

Sp

µ(θH |e)rH+µ(θL|e)rL−rLθL

≻+

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Let’s see what the InC does to the set of separating e’qa. Recallthat rL = wL and rH = wH :

◮ for each e > e∗H , the utility the low type θL gets, is belowwhat he gets if he exerts e = 0

◮ but then all effort levels e > e∗H should be ascribed to the hightype θH and the probability of coming from the low typeshould be set to zero

◮ since the same argument applies for any e > e∗H , there is nopoint in wasting effort on the useless e and the only e’qm noteliminated is the one at the minimum cost e∗H (the so called‘Riley outcome’).

This argument selects a unique separating e’qm!

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e

w

w∗H = rH

wP = E[r ]

eP

u0(θH)u0(θL)

e∗H e1

u′(θH)

u′(θL)

≻+

The InC destroys all e’qa but one sep’g e’qm! 35 / 36




This is great! but

◮ notice that the ‘least-cost’ e’qm is independent ofµi , i ∈ {L,H}, the proportion of skills

◮ suppose now the prior µL gets very small, say, µL = δ → 0

◮ in such a case it seems excessive to pay a high education costof e∗H in order to distinguish the high type from the lowprobability low ability type

◮ just to get a very small increase over the pooling wage.

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