Job market signaling and employer learning · 2012-11-12 · Job market signaling and employer...

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Journal of Economic Theory 147 (2012) 1787–1817www.elsevier.com/locate/jet

Job market signaling and employer learning !

Carlos Alós-Ferrer a,!, Julien Prat b

a University of Konstanz, Department of Economics, Box D-150, D-78457 Konstanz, Germanyb Institute for Economic Analysis (IAE-CSIC), Campus UAB, 08193 Bellaterra, Barcelona, Spain

Received 10 June 2010; final version received 14 December 2011; accepted 30 December 2011

Available online 25 January 2012

Abstract

We consider a signaling model where the sender’s continuation value after signaling depends on his type,for instance because the receiver is able to update his posterior belief. As a leading example, we intro-duce Bayesian learning in a variety of environments ranging from simple two-period to continuous-timemodels with stochastic production. Signaling equilibria present two major departures from those obtainedin models without learning. First, new mixed-strategy equilibria involving multiple pooling are possible.Second, pooling equilibria can survive the Intuitive Criterion when learning is efficient enough.! 2012 Elsevier Inc. All rights reserved.

JEL classification: C72; J24

Keywords: Employer learning; Signaling games; Intuitive Criterion; Multiple pooling

1. Introduction

Signaling games are models for situations where an agent is able to send messages aboutinformation that he could not otherwise credibly reveal. Private information is valuable to the

! We are grateful to two anonymous referees, the associate editor, Egbert Dierker, Ulrich Kamecke, Gerhard Orosel,Larry Samuelson, Gerhard Sorger, seminar participants in Alicante, Ankara, Berlin, Brussels, Hamburg, Konstanz,Maastricht, Valencia, and Vienna, and participants at the 2007 European Economic Association and the 2007 Societyof Economic Dynamics meetings for helpful comments. Julien Prat acknowledges the support of the Barcelona GSE andthe Government of Catalonia.

* Corresponding author. Fax: +49 7531 88 4119.E-mail addresses: [email protected] (C. Alós-Ferrer), [email protected] (J. Prat).

0022-0531/$ – see front matter ! 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2012.01.018

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1788 C. Alós-Ferrer, J. Prat / Journal of Economic Theory 147 (2012) 1787–1817

extent that it helps to predict the outcome of the transaction between the sender and the receiver.When the relationship involves repeated interactions, observing each outcome allows the receiverto update his belief. He is therefore able to gather knowledge about the sender after the signalingstage, a possibility that is usually excluded in signaling games. A direct consequence is thatexpectations about future payoffs do not solely depend on the equilibrium belief, as is commonlyassumed in the literature, but also on the sender’s type. Even when agents are indistinguishableright after the signaling stage, their payoffs vary with their type because they know that it will berevealed over time. The standard premise that senders are equally rewarded in pooling equilibriais therefore violated.

We illustrate this general mechanism and its ramifications in a job-market signaling modelalong the lines of Spence’s [28] seminal work, where workers are better informed about theirability than prospective employers. We embed Spence’s original game into a framework whereworkers’ payoffs depend not only on their signal but also on their type. For concreteness, wewill often focus on employer learning, i.e. on models where employers are able to update theirbeliefs on workers’ types, typically on the basis of observed production. Our approach, however,is more general. It covers any situation where the continuation value after signaling depends onthe sender’s type, for example if receivers have access to a grading technology as in Feltovichet al. [17] or Daley and Green [12].

In signaling models, talented workers have an incentive to invest in education. In our setting,they may instead decide to save on educational costs and trust that their actual productivity willbe revealed by their performance on-the-job. Learning yields higher asset values for talentedindividuals even when pooling is the equilibrium outcome. The gap increases with the efficiencyof the updating process as low and high types become less and more optimistic, respectively. Thestronger the correlation between a worker’s ability and his observable performance, the moreattractive it is for high types to reveal their ability on-the-job instead of paying the educationalcosts.1

Spence’s model does not take into account this countervailing incentive because it assumesthat all information is collected prior to labor market entry. Most of the ensuing theoretical lit-erature follows Spence’s approach, ignoring employer learning.2 Econometricians, on the otherhand, have devised ingenious tests for unobservable characteristics in order to estimate the ef-ficiency of signal extraction in labor markets. Empirical evidence documented in Lange [23]shows that employers are able to elicit information about workers’ abilities quite quickly, withnearly 95% of the statistically significant information being collected after solely 3 years.3

This finding suggests that it is important to assess whether the learning process affects theoutcomes of the signaling game. We show in this paper that this is indeed the case. Our analysisidentifies three important consequences. The first is that the set of pooling equilibria depends on

1 On the other hand, the incentive for low types to send misleading signals, and thus to acquire education, increaseswith the speed of employer learning. Habermalz [19] proposes a partial equilibrium model which underlies the ambiguityof the relationship between the value of job market signaling and the speed of employer learning.

2 A notable exception is Gibbons and Katz [18], who focus on the implications of asymmetric information for layoffdecisions. Feltovich et al. [17] consider a set-up with three types where information is revealed after the signaling stageand show that high and low types may pool at the lowest level of education. More recently, Voorneveld and Weibull [29]study markets for lemons in which buyers receive a private noisy signal of the product’s quality.

3 See also Section 3.4. Available measures of employer learning do not distinguish between symmetric and asymmetriclearning. This is why Pinkston [26] can use an approach similar to that of Lange [23] to estimate a model of asymmetriclearning in which all firms receive a private signal but the current employer always accumulate more information aboutthe worker.

C. Alós-Ferrer, J. Prat / Journal of Economic Theory 147 (2012) 1787–1817 1789

the quality of signal extraction. The second is that qualitatively new types of equilibria may arisewhere more than one common level of education is acquired by both types of workers (multiplepooling). The third consequence concerns forward induction. The multiplicity of equilibria inSpence’s model has been a motivation for the vast literature on refinement concepts. For signalinggames with only two types, the Intuitive Criterion of Cho and Kreps [9] is the most commonlyused refinement because it excludes all but one (separating) equilibrium. This is no longer truein our setting. We show that the forward-induction argument embodied in the Intuitive Criterionloses its predictive power when employer learning is sufficiently fast.

To underline the generality of the results, we initially adopt a reduced-form approach. Wedefine workers’ value functions and specify generic correlations between signaling and workers’payoffs. We show later that these properties are fulfilled in models of passive learning where theworker’s only signal is education and firms update their beliefs using Bayes’ rule, so that typesare gradually revealed. Under this interpretation, our set-up bears similarities to Jovanovic’s [20]matching model with the crucial difference that uncertainty is not match-specific but worker-specific. Hence our examples bridge the gap between the theoretic literature on signaling gamesand the labor market literature.

We view our work as a parsimonious step towards making signaling models more realistic ina direction whose importance has been widely documented in the labor economics literature. Itis our hope that our previously unforeseen findings will pave the way for a systematic programof research on the interactions between learning and signaling. We do not attempt, however,to characterize all the implications of signal extraction because it is our conviction that sucha task is beyond the scope of any paper. We also do not share the objective of the “refinementprogram” by aiming at uniqueness results under powerful refinement concepts. In concurrentand independent research, Daley and Green [12] have achieved progress in that direction. Theirpaper analyzes a signaling game with grades that is similar to the Reports Model presented inSection 2. They show that the D1 criterion restores uniqueness when an additional condition,relating the signaling costs to the precision of the grades, is fulfilled. We discuss some of Daleyand Green’s findings in Section 4.

The paper is structured as follows. Section 2 lays out the model’s set-up and provides anillustrative example (the Reports Model). Section 3 analyzes the equilibria in pure and mixedstrategies and outlines some implications for empirical research. Section 4 discusses the condi-tions under which the Intuitive Criterion can refine the set of equilibria and explains why theyare not met when employer learning is efficient enough. Section 5 derives the workers’ valuefunctions from first principles in two-period and continuous-time settings. Section 6 concludes.Proofs are relegated to Appendix A.

2. Signaling and employer learning

Workers differ in their innate abilities. They can be of different types which determine theirproductivity. Nature initially selects types according to pre-specified probabilities. The main in-gredient of the model is information asymmetry: Workers know their abilities whereas employersmust infer them.

The game is as follows. In a first step, the worker chooses an education level. In a secondstep, the industry offers a starting wage based on beliefs derived from the education signal. InSpence’s static framework, the game ends as the worker enters the labor market. In our set-up, incontrast, we consider a third step where the relationship develops, with the industry being able toextract information from noisy realizations of the worker’s productivity. Rather than proposing


a particular dynamic model, we adopt a reduced-form approach and delay its microfoundation toSection 5.

I. Education decision. We restrict our attention to cases where workers (senders) are of only twotypes, i = h (high) or i = l (low). Nature assigns a productivity p " {pl,ph} to the worker, withph > pl . High types account for a share µ0 " (0,1) of the population. For simplicity, we assumethat workers are infinitely lived and that they discount the future at the common rate r . Beforeentering the labor market, workers choose their education level e " [0,+#). To isolate the effectof signaling, we do not allow education to increase productivity. Its only use is to signal abilitieswhich are initially unobserved by the industry (receiver).

Let the function c :R++ $R+ % R+ specify the cost of acquiring education. That is, c(p, e)

is the cost to acquire education e for a worker with productivity p. The cost function is twicedifferentiable with ce(p, e) > 0 and cee(p, e) ! 0, hence strictly increasing and convex in thelevel of education. As commonly assumed, we also let cp(p, e) < 0 and cpe(p, e) < 0. The lastrequirement ensures that the Single-Crossing Property is satisfied because low types have steeperindifference curves.

II. Wage setting. As in Spence’s model, a worker is paid his expected productivity. This wage set-ting rule is justified as a proxy for a competitive labor market or a finite number of firms engagedin Bertrand competition for the services of the worker.4 Let µ(e), which we abbreviate by µ whenno confusion may arise, denote the probability that the industry attaches to the worker being ofthe high type given education signal e. The initial wage is given by w(µ) = (1 & µ)pl + µph.We can therefore use the equilibrium belief of the industry to denote its response to a particularsignal.

III. Expected income. The key departure from Spence’s model is the third step where we specifythe sender’s expected income. We adopt in this section a reduced-form approach: Given theactual productivity p and the employer’s prior µ = µ(e), expected earnings are given by a valuefunction v(p,µ) which captures the impact that correlated information has on future income.

For the purposes of enabling comparative statics, we explicitly introduce two additional pa-rameters. The first one is the common discount rate r of workers. We adopt the convention thatthe forward-discounted value is given by w/r when the agent receives a constant wage w, as ina continuous-time model with r > 0. The second parameter, s > 0, measures the informativenessor precision of additional, post-education signals on worker’s productivity. Thus we will writev(p,µ|r, s) whenever we wish to discuss the effect of those two parameters.

Let us now specify the properties that should be imposed on value functions and the ratio-nales behind them. As employment histories unfold, employers observe the cumulative outputsproduced by workers and use this information to revise their priors. Consider how the updat-ing process affects the expectations of low types. For every µ " (0,1), they are offered an initialwage w(µ) that is above their actual productivity pl . On average, realized outputs will induce theindustry to lower its belief and thus wages. Their expected income v(pl,µ) is therefore smallerthan the forward-discounted value of the starting wage w(µ)/r , but larger than the value pl/r

that they would have obtained if the industry had known their type with certainty. A symmetric

4 The first approach can be formalized by assuming a single receiver (“the industry”) with payoffs given by a quadraticloss function &(w & p)2. If the latter approach is adopted, the equilibrium concept must be refined to ensure that allfirms share the same beliefs. See Section 13.C of the textbook by Mas-Colell et al. [25] for a discussion.


argument holds for high types. On the other hand, when industry’s beliefs collapse to certainty,that is µ " {0,1}, further information will be ignored and the initial wage will never be altered.Since we want to encompass Spence’s model as a particular case, we first introduce a weakimplication of this argument:

P0. For all µ " (0,1),

ph

r> v(ph,µ) ! w(µ)

r! v(pl,µ) >

pl

r.

Further, when µ " {0,1}, v(ph,µ) = v(pl,µ) = w(µ)/r .

Because of learning, we actually expect all inequalities to be strict. This stronger variant of P0ensures that firms are able to update their beliefs towards the true productivity.

P1. For all µ " (0,1),

ph

r> v(ph,µ) >

w(µ)

r> v(pl,µ) >

pl

r.

Further, when µ " {0,1}, v(ph,µ) = v(pl,µ) = w(µ)/r .

Compare now two industries whose correlated signals have different precision. Workerswhose productivity is overestimated prefer being employed in the industry where signal extrac-tion is slow. Conversely, underestimated workers would rather work in the industry where actualabilities are quickly recognized. The following property captures this intuition.

P2. For all µ " (0,1), we have that !v(ph,µ|r, s)/!s > 0 and !v(pl,µ|r, s)/!s < 0, hencev(ph,µ|r, s) is strictly increasing in s whereas v(pl,µ|r, s) is strictly decreasing in s.

Finally, consider limit cases where precision is very low or very high. As s goes to zero,employers have no possibility to update their initial beliefs: Wages are never revised and thevalue functions converge to their original specification in Spence’s model. Conversely, when sgoes to infinity, types are immediately recognized and signaling becomes redundant.

P3. For all µ " (0,1), lims%0 v(p,µ|r, s) = w(µ)/r and lims%# v(p,µ|r, s) = p/r .

Definition 2.1. A value function is a mapping assigning a lifetime income v(p,µ) to eachbelief µ " [0,1] and productivity p " {pl,ph}. It is assumed to be strictly increasing and twicedifferentiable in µ, and to fulfill property P0.

A value function is said to exhibit weak learning if it fulfills the more demanding property P1.It exhibits strong learning if it fulfills properties P1, P2, and P3.

Spence’s problem is encompassed in our framework when the value function v(p,µ) =w(µ)/r , with the interpretation that the starting wage is final and no further adjustments (ofwages or beliefs) are possible. This function fulfills the basic conditions in Definition 2.1. Alter-natively, it can be interpreted as a limit case of a value function with strong learning when s goesto 0.

The stronger properties listed in Definition 2.1 capture the basic intuition on employer learn-ing. We relegate the dynamic microfoundations of those properties to Section 5, where we


describe a setting with explicit learning based on actual production. We will also illustrate therehow to derive s from primitive parameters. In order to clarify the exposition, we present now anillustrative model where the postulated properties follow from Bayesian learning when employ-ers receive an additional, informative but imprecise signal on workers’ productivity.

IV. The Reports Model. We assume that updating occurs only once over the labor market career ofa given worker. The employer has access to a detection technology which delivers a signal aboutthe agent’s type. We call this signal a “report” to avoid confusion with the education decision.The report is extracted as soon as the agent starts working. One may think of the report asresulting from the worker’s performance during his job interview, a trial period, etc. There aretwo possible reports, G(ood) and B(ad). If the productivity of the worker is high, a good reportis delivered with probability d(s) > 1/2 for s " (0,+#), and a bad report is delivered with thecomplementary probability. When the worker’s productivity is low, the likelihoods of a good orbad report are 1 & d(s) < 1

2 and d(s), respectively. The function d(s) is strictly increasing in swith the following lower and upper bounds: lims%0 d(s) = 1/2 and lims%# d(s) = 1. Hence theparameter s measures the informativeness of the post-education signal. Since d is a change ofvariable, we simply write d = d(s).

Firms use reports to update their beliefs and then pay workers their expected productivity overan infinite horizon. Thus the expected lifetime income of an employee with ability i is simply

v(p,µ|r, s) = 1rE

!w

"µ'#|p,µ, s

$, (1)

where µ' denotes the updated belief of the industry and E[·|p,µ, s] is the expectation operatorconditional on the industry’s prior, worker’s type and precision of the correlated information s.

As the updated belief µ' is formed according to Bayes’ rule, we have

µ'(µ,G) = µd

µd + (1 & µ)(1 & d)and µ'(µ,B) = µ(1 & d)

µ(1 & d) + (1 & µ)d.

Wages w(µ,S) = w(µ'(µ,S)) therefore depend on the signal S " {B,G} in the following way

w(µ,G) = µdph + (1 & µ)(1 & d)pl

µd + (1 & µ)(1 & d)and w(µ,B) = µ(1 & d)ph + (1 & µ)dpl

µ(1 & d) + (1 & µ)d.

The expected wage in the second period reads

E!w

"µ'#|pl,µ

$= (1 & d)w(µ,G) + dw(µ,B) and

E!w

"µ'#|ph,µ

$= dw(µ,G) + (1 & d)w(µ,B).

Straightforward but cumbersome computations show that v(p,µ|r, s) fulfills all the requirementsfor strong learning listed in Definition 2.1.

An alternative, slightly more involved model would be obtained if firms cannot immediatelyupdate their beliefs because reports are delivered e.g. at the end of a first period of production. Insuch a “two-period Reports Model”, the expected lifetime income of an employee with ability iis given by5

v(p,µ|r, s) = 11 + r

%w(µ) + E[w(µ')|p,µ, s]

r

&. (2)

5 The scale factor 1/(1 + r) is immaterial to the analysis and has been introduced for consistency across specificationssince it ensures that receiving the wage w forever yields a value of w/r .


The value function now displays weak and not strong learning, because lims%# v(p,µ|r, s) (=p/r , hence property P3 fails. The reason is that employer learning matters solely for earnings af-ter the non-negligible first period. Whether the model generates weak or strong learning cruciallydepends on the timing of the report. For example, in a model with infinitely many production pe-riods (e.g. production in continuous time), initial wages become negligible in the limit, and stronglearning is restored (see Section 5). The contrast between the one- and two-period versions of theReports Model shows, however, that this is not due to the difference between finite and infinitehorizon, but to whether or not beliefs before the first additional signal have a negligible impacton lifetime income.

V. Equilibrium concept. The equilibrium concept is just Perfect Bayesian Equilibrium for thegame as specified before.6 Combining the value and cost functions yields the payoff function forworkers: u(e,µ|p) " v(p,µ) & c(p, e). As discussed in paragraph II, industry’s payoffs are in-consequential as long as the specification leads to wage offers equal to the expected productivitygiven the beliefs.

We will consider equilibria in pure and mixed strategies (“hybrid equilibria”). The very for-malization of both Spence’s model and ours is such that the industry never has an incentive torandomize. Thus an equilibrium in mixed strategies can be defined as follows.

A signaling equilibrium is made out of a pair of probability distributions (q(·|p))p"{pl,ph}on R+ describing the education levels chosen by both types, and a belief system (µ(e))e"R+

describing the priors of the industry given any possible signal, that satisfies the following prop-erties:

• Sequential rationality for the workers: For p " {pl,ph}, if e! belongs to the support ofq(·|p), then e! " arg maxe"R+ u(e,µ(e)|p).

• (Weak) consistency of beliefs: The industry’s initial beliefs µ(e) are consistent with Bayes’rule for any educational attainment e in the support of either q(·|ph) or q(·|pl).

• Sequential rationality for the industry: Given any education level e, the industry offers aninitial wage equal to w(µ(e)).

In the pure strategy case, the support of the distributions q(·|p) is a singleton, implyingthat a signaling equilibrium in pure strategies can be described by a belief system as before,and a strategy profile (el, eh) describing the deterministic education levels chosen by bothtypes.7

3. Equilibrium analysis

We proceed now to characterize the signaling equilibria of the game. We start with pure-strategy equilibria, which are fully described by the beliefs µ(e) and the education levels selectedby both types, el and eh. In the following subsections, we characterize mixed-strategy equilibriaand then illustrate them in the framework of the Reports Model. The last subsection discussessome empirical implications.

6 Some of the earlier literature called PBEs “sequential” in reference to the sequential rationality requirement. Theconcept of sequential equilibrium (Kreps and Wilson [22]), however, is defined for finite games only.

7 Workers are required to randomize among optimal education levels only. This eliminates technical difficulties asso-ciated with the inclusion of suboptimal strategies with zero probability in the support of the equilibrium strategies.


3.1. Pure-strategy equilibria

Separating equilibria are such that abilities are perfectly revealed: Depending on the educationsignal, the beliefs of the industry are either µ = 0 or µ = 1. Each type receives an initial wageequal to his productivity and P1 implies that v(pi,1) = pi/r for i " {l, h}. In other words, em-ployer learning does not affect separating equilibria and the following proposition follows fromstandard arguments.

Proposition 1. Consider any value function. An education profile (el, eh) with el (= eh can besustained as a (separating) signaling equilibrium if and only if (i) el = 0; and (ii) eh " [eh, eh]where eh > eh > 0 and these two education levels are uniquely defined by

c(pl, eh) & c(pl,0) = 1r(ph & pl) = c(ph, eh) & c(ph,0).

In pooling equilibria, both types select the same education level ep and are therefore offeredthe same initial wage w(µ0). Yet they have different expected incomes as v(ph,µ0) > v(pl,µ0).This implies that the set of pooling equilibria shrinks as information precision increases.

Proposition 2. Consider any value function. A common education level ep can be sustainedas a pooling equilibrium if and only if ep " [0, ep] where ep > 0 is uniquely defined by theequation

c(pl, ep) & c(pl,0) = v(pl,µ0) & pl

r.

The upper bound fulfills ep < eh, where eh is as defined in Proposition 1. Further, under stronglearning, ep is strictly decreasing in the precision s and lims%# ep = 0.

The set of pooling equilibria is a subset of the one in Spence’s model. Learning lowers theupper bound ep because the incentives for low types to mimic high types decrease as the abilityof firms to detect them improve. Under strong learning and as informativeness s goes to infinity,firms ignore education levels in favor of information collected after the signaling stage. Thereis no reason to spend resources on signaling, and the set of pooling equilibria collapses to theminimum education level.

3.2. Mixed-strategy equilibria

As usual in signaling games, the model admits a plethora of mixed-strategy equilibria, of-ten referred to as hybrid equilibria. The following proposition shows that learning supportsmixed-strategy equilibria which are qualitatively different from those possible in Spence’smodel.

Proposition 3. Consider any value function. For any mixed-strategy signaling equilibrium, thereexists a set of education levels Ep ) [0, eh], where eh is as defined in Proposition 1, such that:

(i) The support of the low and high types’ strategies consists of Ep and at most a further edu-cation level per type, el = 0 and eh respectively, such that el < e < eh for all e " Ep .

(ii) The industry’s beliefs µ are strictly increasing on Ep .


Equilibria with Ep = * are the separating equilibria of Proposition 1. The pooling equilibriaof Proposition 2 are such that Ep = {ep} where no additional education levels are chosen. InSpence’s model, the set Ep (if nonempty) always consists of a unique education level, but thisis not true in general for other value functions. Correlated information qualitatively enriches theset of outcomes: multiple pooling may arise, i.e. equilibria with more than one education levelchosen by both types.

Although the set of mixed-strategy equilibria might be large and capture complex phenomenaas multiple pooling, such equilibria are far from being arbitrary. In particular, the monotonic-ity property (ii) shows that there can be no counterintuitive correlation: Education remainsinformative in the sense that more education is always associated with a higher average pro-ductivity.8

To better understand why multiple pooling equilibria may arise, consider the model withoutcorrelated information. If a worker chooses two different levels of education with positive proba-bility, he must be indifferent between those two education levels, hence the difference in expectedincome must equal that in educational costs. Without employer learning, each level of educationresults in identical incomes for both types. By the single-crossing property, however, differencesin educational costs across signals are higher for low types. Hence both types cannot randomizeover the same set.

The contradiction disappears with employer learning because, even when priors are identical,lifetime earnings differ across types. Then the difference in the increase in education costs re-quired by the single-crossing property can be compensated by different gains in lifetime earningsacross types.

We remark that the set of equilibria is also enriched if one moves away from the classicalsignaling setting by relaxing the single-crossing property. Then, the contradiction mentionedabove does not follow because differences in educational costs across signals can be identical forboth types. Araujo and Moreira [4] establish that a violation of the single-crossing property maylead to continuous pooling in adverse selection problems. It would be interesting to apply theirmethodology to signaling models in order to distinguish the impact of the signaling technologyfrom that of the learning process.

3.3. Multiple pooling equilibria with linear costs

We now present an extended example to better understand multiple pooling equilibria. Weconsider the Reports Model with linear cost functions c(p, e) = g(p)e, where g'(p) < 0. Sup-pose that a mixed-strategy equilibrium exists and let e1, e2 " Ep with e1 < e2. By definition, bothtypes have to be indifferent between e1 and e2, i.e.

v"pi,µ(e1)

#& g(pi)e1 = v

"pi,µ(e2)

#& g(pi)e2, (3)

for i " {l, h}. We define the function

h(µ) " v(ph,µ) & g(ph)

g(pl)v(pl,µ). (4)

8 Proposition 3 can be refined assuming that, for all µ " (0,1), !2v(ph,µ)/!µ2 < !2v(pl,µ)/!µ2, a property thatcan easily be established in the Reports Model. Then, P1 implies that there exists a unique belief µ! " (0,1) such that!v(pl,µ

!)/!µ # !v(ph,µ!)/!µ if and only if µ # µ!. It is then straightforward to show that, under weak or stronglearning, µ(e) > µ! for all e " Ep except possibly the minimum education level in Ep (if a minimum exists).


Fig. 1. Beliefs supporting mixed-strategy equilibria. Parameters: pl = 0, ph = 1, g(pl) = 0.15, g(ph) = 0.1, d = 3/4,r = 1.

Combining the indifference condition (3) for both types, it is easily seen that a necessary con-dition for the existence of mixed-strategy equilibria is that there exist µ1,µ2 " [0,1], withµ1 (= µ2, such that

h(µ1) = h(µ2). (5)

Fig. 1 depicts the function h(µ) for a particular set of parameters. It can be shown that h isstrictly concave in general. Hence, condition (5) can only hold true for a pair of beliefs and theset Ep contains at most two signals.

The set of PBE crucially depends on the share µ0 of high types in the population. Let µmin < 1be the unique interior belief where h(µmin) = h(1), as in Fig. 1. If µ0 > µmin, we can picka pair of beliefs {µ1,µ2} such that µ0 " (µ1,µ2) and h(µ1) = h(µ2). Then we can constructa “bipolar” equilibrium where neither type separates by sending a third education level. Theequilibrium randomization strategies q(e|p) are uniquely pinned down by weak consistency ofbeliefs, because q(e1|pl) and q(e1|ph) deliver two linear equations and two unknowns. Theparticipation constraint of the low types9 delivers an upper bound for the value of e1 (while e2is uniquely determined from e1 by indifference), resulting in a set of possible multiple poolingequilibria for the selected µ1, µ2.

Suppose now that µ0 < µmin, implying µ0 < µ1 < µ2. This holds in equilibrium if and onlyif low types randomize on Ep and the additional separating signal e0 = 0. “Bipolar” equilibriacannot anymore be sequentially rational. Consistent randomization strategies can neverthelessbe easily constructed for appropriate values of q(0|pl), because one has an additional degree offreedom. Hence there exists a continuum of randomization strategies associated to any beliefs.The education levels, by contrast, are uniquely determined: Given that low types also separate

9 As usual in signaling problems, the high types’ constraint is redundant due to the single-crossing property.


by acquiring zero education, their participation constraint u(e1,µ1|pl) ! pl/r will bind, pinningdown the value of e1.

This heuristic reasoning suggests a characterization of the set of equilibria. The followingproposition describes its actual features and provides an explicit, necessary and sufficient condi-tion for the existence of multiple pooling equilibria.

Proposition 4. Consider the Reports Model with linear costs, c(p, e) = g(p)e and 0 <g(ph) < g(pl). Multiple pooling equilibria exist if and only if

d >12

'1 +

(g(pl) & g(ph)

g(pl) + 3g(ph)

). (6)

(a) In any multiple pooling equilibrium, both types randomize on a common set containingexactly two signals, Ep = {e1, e2} with 0 < e1 < e2. Moreover, high types randomize ex-clusively among those two signals, i.e. there is no additional separating signal for hightypes.

(b) There always exist multiple pooling equilibria where low types randomize on Ep and anadditional separating signal e0 = 0.

(c) There exists a µmin such that “bipolar” equilibria, where the low types randomize exclu-sively among the two signals in Ep , may arise if (and only if ) µ0 > µmin.

As expected, pooling over more than one signal is ruled out when there is no correlated infor-mation (d = 1/2). Conversely, reducing the gap in educational costs between both types enlargesthe parameter set over which mixed strategies are sequentially rational. This is because differ-ences in the slope of the value functions have to offset those in educational costs. As the latteris reduced, less requirements are imposed on the value functions. In the limit case where thetwo cost functions converge, g(pl) % g(ph), the necessary and sufficient condition (6) is alwaysfulfilled.

3.4. Empirical implications of multiple pooling

The statistical discrimination literature, e.g. Farber and Gibbons [16] or Altonji and Pier-ret [2], shows that firms update their beliefs about the ability of their employees. The identifyingassumption is that econometricians have access to a correlate of abilities that is not availableto employers, usually Armed Forces Qualification Test scores of workers. The impact of thiscorrelate on wages increases with labor market experience, indicating that firms do learn overtime.

This finding suggests that relevant information remains hidden because some signals are sentby both types. In pure pooling equilibria, education does not convey any information. The up-dating process and the wage dynamics are independent of the educational choice. They dependsolely on the informativeness of the production technology. In contrast, the multiple poolingequilibria described in Section 3.2 restore a connection between labor market and educationaloutcomes. Although the returns to education for each type are pinned down by their cost func-tions, they cannot be identified in the data because one observes a pool of workers with a givenlevel of education. Observable returns are therefore affected by a composition effect.10 It is most

10 We thank an anonymous referee for bringing this mechanism to our attention.


easily characterized in the Reports Model with linear costs. By definition, observable returns aregiven by (E[v|e] & E[v|0])/e with

E[v|e] = µ(e)v"ph,µ(e)

#+

"1 & µ(e)

#v"pl,µ(e)

#.

Consider situations where precision is high enough for Eq. (6) to hold. Then workers may ran-domize over three education levels and the indifference requirement for low types implies thatv(pl,µ(e)) & pl/r = g(pl)e. Thus observable returns are equal to

E[v|e] & E[v|0]e

= g(pl) + µ(e)v(ph,µ(e)) & v(pl,µ(e))

e.

The second term captures the composition effect, showing that it depends on two channels. First,high types account for a larger share of workers as education, and consequently µ, increase.This raises the education premium because high types have higher expected earnings than lowtypes. On the other hand, high types also have lower marginal costs and so the indifferencerequirement across signals implies that the gap in expected earnings decreases with education.Whether observable returns are positively correlated with education depends on which of thesetwo channels dominates. In most configurations, the second channel turns out to be stronger sothat the expression above decreases in e.11 This negative composition effect may help explainempirical evidence about the slim premium commanded by PhDs over master’s degrees (seeCasey [8] for recent evidence).

4. The Intuitive Criterion

Even without correlated information, signaling games typically have a large set of equilibria.In order to narrow it, a number of refinement concepts have been developed. Yet the IntuitiveCriterion (Cho and Kreps [9]) remains a milestone in the analysis of signaling games: Whenthere are only two types, it confers a predictive power to Spence’s model by ruling out all but oneseparating equilibrium, known as the Riley equilibrium. The purpose of this section is to showthat this does not hold true when beliefs can be updated after the signaling stage. More precisely,we prove that, even though the Riley equilibrium retains its importance as the only separatingequilibrium fulfilling the Intuitive Criterion, it is in general not true that all pooling (or mixed)equilibria are ruled out.

A signaling equilibrium is said to fail the Intuitive Criterion if some type could strictly profitby sending a non-equilibrium signal, provided that the sender adopts non-equilibrium beliefs as-signing probability zero to types which could never conceivably profit by sending the consideredsignal. In our set-up, the Intuitive Criterion amounts to the following. Fix a signaling equilib-rium. Say that an unused signal e is equilibrium-dominated for type p if the equilibrium payoffof type p is strictly larger than the payoff that type p would receive with signal e, given anyconceivable (non-equilibrium) belief of the industry µ'(e). The equilibrium is said to fail the In-tuitive Criterion if there exists a signal e and a type p such that the equilibrium payoff of type p

is strictly smaller than the minimum payoff that this type could get by sending signal e, givenany possible (non-equilibrium) beliefs µ'(e) of the industry which concentrate on the set of typesfor which signal e is not equilibrium-dominated.

11 The composition effect can be shown to be negative if g(pl)e/µ(e) = (v(pl,µ(e)) & v(pl,0))/µ(e) increasesin e.


4.1. Separating equilibria

In the absence of employer learning, the Riley equilibrium is the only separating equilibriumwhich survives the Intuitive Criterion: Low types do not acquire any education and high typessend the signal eh defined in Proposition 1, implying that low types are indifferent between theirequilibrium strategy and acquiring education eh in order to receive v(pl,1) = ph/r . Given thatseparating equilibria do not depend on the value function, it is not surprising that the result carriesover to our set-up. The proof follows from standard arguments.

Proposition 5. Consider any value function. The only separating equilibrium which survives theIntuitive Criterion is the Riley equilibrium.

4.2. Survival of pooling equilibria

We now focus on pooling equilibria and use ep to denote the common level of education. First,notice that no signal e < ep can be equilibrium-dominated for either type. Further, if a deviationto e > ep is equilibrium-dominated for the high type, it is never profitable for the low type (in thesense of the Intuitive Criterion) to choose e. For such a deviation would induce industry’s beliefsµ(e) = 0 and low types would obtain lower payoffs (pl/r) than at the pooling equilibrium butincur strictly larger educational costs.

Hence, a pooling equilibrium with education level ep fails the Intuitive Criterion if and onlyif there exists a signal e > ep such that it is equilibrium-dominated for the low types but wouldresult in a better payoff than in equilibrium for the high types when the industry places zeroprobability on the event that the sender is of the low type given signal e, i.e.

ph

r& c(pl, e) < v(pl,µ0) & c(pl, ep) (equilibrium dominance for the low type), (ED)

ph

r& c(ph, e) > v(ph,µ0) & c(ph, ep) (profitable deviation for the high type). (PD)

The equilibrium dominance condition (ED) implies that, even in the best-case scenario wherethe worker could forever deceive employers, deviating to e is not attractive to low types. Thefirm can therefore infer by forward induction that any worker with an off-equilibrium signal ehas a high productivity. The “profitable deviation” condition (PD) implies in turn that crediblydeviating to e is indeed profitable for the high type. Thus such an ep fails the Intuitive Criterion.

Let e!(ep) denote the minimum education level that does not trigger a profitable deviationfor low ability workers, so that (ED) holds with equality at e!(ep). Analogously, let e!!(ep)be such that (PD) holds with equality. These two thresholds always exist because c(p, e) iscontinuous and strictly increasing in e.12 Conditions (ED) and (PD) can then be rewritten ase!(ep) < e < e!!(ep). Such an e exists if and only if e!(ep) < e!!(ep). This condition providesus with a straightforward proof that all pooling equilibria fail the Intuitive Criterion in the modelwithout learning. Assume that ep is stable, so that condition (PD) is not satisfied at e!(ep). Thiscan be true if and only if

v(ph,µ0) & v(pl,µ0) ! c"pl, e

!(ep)#& c(pl, ep) &

!c"ph, e

!(ep)#& c(ph, ep)

$. (7)

12 To see this, observe first that, because v(pl,µ0) < ph/r , we have ph/r & c(ph, ep) > v(ph,µ0) & c(ph, e) fore = ep . Further, the cost function c(p, e) is continuous, strictly increasing, and convex in e, hence the inequality isreversed for e large enough. This implies that e!(ep) and e!!(ep) are well defined. It also follows that e!(ep) > ep ande!!(ep) > ep .


Fig. 2. Workers’ indifference curves.

In Spence’s model or as s % 0, we have v(pl,µ0) = v(ph,µ0) = w(µ0)/r . The left-hand sideof inequality (7) converges to zero while the right-hand side is strictly positive by the single-crossing property. The contradiction illustrates that, in the basic signaling model, one can alwaysfind a credible and profitable deviation for high types.

When workers’ abilities are also revealed on-the-job, the premise leading to a contradictionis no longer valid. As stated in property P1, v(ph,µ0|r, s) > v(pl,µ0|r, s) for all s > 0, and soinequality (7) can hold true for some parameter configurations.

While it is easy to construct numerical examples showing that pooling equilibria might survivethe Intuitive Criterion, we can also provide general results. According to property P2, the moreprecise correlated information is, the wider the gap in expected income between low and hightypes. This suggests that a pooling equilibrium is more likely to be stable when signal extractionis efficient. The following proposition substantiates this intuition, showing that any pooling equi-librium is stable when precision is high enough. For the particular case of linear costs, we findan even sharper result.

Proposition 6. The Intuitive Criterion rules out all pooling equilibria in the absence of learning,but this is no longer true for value functions with either weak or strong learning. Further, for anyvalue function with strong learning, the following hold:

(a) For any education level ep , there exists a precision s!(ep) such that, for any s ! s!(ep), ifep can be sustained as a pooling equilibrium then it survives the Intuitive Criterion.

(b) If costs are linear, there exists a precision s! such that, for any s ! s!, every pooling equi-librium survives the Intuitive Criterion.

Fig. 2 illustrates the mechanism behind Proposition 6. It displays the indifference curves ofhigh and low types when s = s!(ep) and when s = 0. The dotted curves correspond to the for-mer case, the undotted curves to the latter one, that is, Spence’s model. The level of education


e!(ep|s) is given by the point where the indifference curve of low types crosses the horizontalline with intercept ph/r . Similarly, the level of education e!!(ep|s) is given by the point wherethe indifference curve of high types crosses the same horizontal line. The pooling equilibrium ep

fails the Intuitive Criterion if and only if e!(ep) < e!!(ep). We can therefore conclude that ep isnot stable when the indifference curve of low types intersects the horizontal line with interceptph/r before the indifference curve of high types.

Consider first the basic model without learning. At the pooling level of education ep , the twotypes enjoy the same asset value w(µ0)/r . The single-crossing property implies that e!(ep) liesto the left of e!!(ep), as shown in Fig. 2. Thus any pooling equilibrium fails the Intuitive Criterionwhen there is no learning. Consider now what happens when the precision s increases. As hightypes are more quickly recognized, their asset value increases and their indifference curve shiftsup. Conversely, the indifference curve of low types shifts down. These opposite adjustmentsshrink the gap between e!(ep|s) and e!!(ep|s). The threshold precision s!(ep) is identified bythe unique point where the gap vanishes as the two indifference curves concurrently cross thehorizontal line with intercept ph/r . Property P3 ensures that one can always find such a pointfor any given ep because lims%# v(ph,µ|r, s) = ph/r .

The economics behind Proposition 6 makes intuitive sense. When learning is fast, firms eas-ily infer the actual type of their employees, and the benefits of ex-ante signaling are small.Conversely, when learning is slow, firms learn little from observed outputs. This leaves feweropportunities for high types to reveal their ability on-the-job and thus raises their incentives tosend a message.

By Proposition 6, when precision is high enough, the Intuitive Criterion does not rule out anypooling equilibrium. The set of pooling equilibria, however, shrinks as precision increases byProposition 2. Hence, there are three possibilities: (i) if learning is slow, there is a large set ofpooling equilibria, almost all (or all) of which fail the Intuitive Criterion; (ii) if learning is fast,pooling equilibria would survive the Intuitive Criterion, but the set of such equilibria is small;and (iii) for intermediate values of s, there is a sizeable set of pooling equilibria surviving theIntuitive Criterion.

This qualitative classification can be made more precise by assuming that the log of the deriva-tive of c(p, e) with respect to e has (strictly) decreasing differences in (p, e) or, in other words,that ce(p, e) is (strictly) log-submodular. Log-submodularity is more stringent than the single-crossing property because it requires marginal educational costs to diverge so that13

ce(ph, e'')

ce(ph, e')>

ce(pl, e'')

ce(pl, e'), whenever e'' > e'.

Proposition 7. Let ce(·) be strictly log-submodular, and consider any value function with stronglearning. Let ep(s) be as in Proposition 2. Then, there exists an s!(0) > 0 and an s > s!(0) suchthat:

(a) For all s " [0, s!(0)[, all pooling equilibria fail the Intuitive Criterion.(b) For all s " [s!(0), s[, there exists *e(s), strictly decreasing in s, such that (i) *e(s) < ep(s),

(ii) all pooling equilibria with education level ep " [0,*e(s)] survive the Intuitive Criterion,

13 This assumption might seem restrictive but it is actually satisfied by the functions commonly used to illustrate thesingle-crossing property. Textbook examples of cost functions usually exhibit log-linear marginal costs which, as statedat the end of Proposition 7, yields an even simpler division of the parameter space.


Fig. 3. Graphical interpretation of Proposition 7. A pooling equilibrium with education level e exists if and only ife $ e(s). It satisfies the Intuitive Criterion if and only if e $*e(s). Parameters for the continuous-time model: r = 0.2,pl = 0.5, ph = 1, µ0 = 0.5 and c(p, e) = exp(e/p) & 1.

and (iii) all pooling equilibria with education level ep " ]*e(s), ep(s)] fail the Intuitive Cri-terion.

(c) For all s ! s, all pooling equilibria survive the Intuitive Criterion.

If ce is log-linear, the result holds with s!(0) = s, i.e. case (b) cannot occur.

This result is illustrated in Fig. 3. Whereas the statement in Proposition 6 is local, assuminglog-submodularity allows for a global characterization of the region where the Intuitive Criterionbites.

4.3. Survival of mixed-strategy equilibria

Mixed-strategy equilibria may survive the Intuitive Criterion for the same reason as poolingequilibria. One simply has to verify that conditions (ED) and (PD) are fulfilled when the share µ0of high types in the population is replaced by the equilibrium belief µ(e). Furthermore, one doesnot need to go through each and every equilibrium signal. As stated in the following proposition,it is sufficient to focus on an arbitrary signal sent by both types.

Proposition 8. A mixed-strategy equilibrium (with Ep (= *) fails the Intuitive Criterion if andonly if for some ep " Ep , there exists an education level e such that

ph

r& c(pl, e) < v

"pl,µ(ep)

#& c(pl, ep) and

ph

r& c(ph, e) > v

"ph,µ(ep)

#& c(ph, ep).

As an example, consider the Reports Model with linear costs as in Section 3.2, and focus onthe mixed-strategy equilibria with three education levels exhibited in Proposition 4. Note that


a failure of the Intuitive Criterion must involve a deviation to a signal e > e2. Then equilibriumdominance for the low types reduces to

pl

r>

ph

r& g(pl)e + e >

1rg(pl)

[ph & pl] > 0,

which provides a lower bound for the deviation e > e2. That deviation would be strictly profitablefor the high types if and only if

v"ph,µ(e2)

#& g(ph)e2 <

ph

r& g(ph)e + ph/r & v(ph,µ(e2))

g(ph)+ e2 > e.

Given that e is bounded below and lims%# v(ph,µ(e2)) = ph/r , both conditions are eventuallyincompatible, showing that the equilibrium survives the Intuitive Criterion.

4.4. Further equilibrium refinements

We have focused on the Intuitive Criterion because of its relevance for Spence’s model. How-ever, many more sophisticated refinement concepts have been proposed and it would be naturalto investigate their predictive power in our setting. Although such a task is beyond the scope ofthis paper, we offer here two observations, focusing on two different avenues: Universal Divinityand the Undefeated Criterion.

The first observation is that existing results based on forward induction do not always helprefine the equilibrium set, so that further research is needed. To substantiate this claim, considerthe classical uniqueness result of Cho and Sobel [10]. That result concerns three particularlyappealing refinement concepts: criterion D1 (Cho and Kreps [9]), Universal Divinity (Banks andSobel [6]), and Never a Weak Best Response (NWBR; Kohlberg and Mertens [21]). It providesa set of necessary conditions ensuring that those three refinements select a unique signalingequilibrium within a subclass of signaling games known as monotonic signaling games (Choand Sobel [10, p. 387]). A monotonic signaling game is such that, for every fixed signal, alltypes have the same preferences over the actions of the receiver. This condition is fulfilled byjob-market signaling games, with or without employer learning.14 Proposition 3.1 in Cho andSobel [10] then states that the three refinements mentioned above coincide, and their main resultestablishes uniqueness.15 The only non-technical condition in their uniqueness theorem is A4,which, in Cho and Sobel’s words, “is crucial to [the] analysis. It states that if two signal-actionpairs yield the same utility to some type of Sender, and one signal is greater (componentwise)than the other, then all higher types prefer to send the greater signal”. The formal condition readsas follows: if p < p' and e < e', then u(e,µ|p)$ u(e',µ'|p) + u(e,µ|p') < u(e',µ'|p').

In Spence’s model, this condition is implied by the single-crossing property, for the wagegiven an education level is independent of the type. Hence, Universal Divinity (or D1) will al-ways select a unique equilibrium. In our setting, however, this condition is not always fulfilled.With employer learning the link fails because lifetime earnings differ across types. Further, it isimmediate that, if a signaling game satisfies condition A4, there cannot be equilibria with multi-ple pooling, for two types cannot be simultaneously indifferent between two signal-action pairswith different signals.

14 In our setting, this is implied by the fact that v(p,µ) is strictly increasing in µ for both p = ph and p = pl .15 Esö and Schummer’s [14] Vulnerability to Credible Deviations provides an alternative interpretation of this selectionresult within the context of monotonic signaling games.


In our view, condition A4 would place strong restrictions on a model with employer learning.With two types, it implies that, whenever v(pl,µ

') & v(pl,µ) ! c(pl, e') & c(pl, e), it must also

hold that v(ph,µ')&v(ph,µ) > c(ph, e

')&c(ph, e). Further, this must be true for all µ, µ', e, e'

with e' > e. When lifetime incomes are independent of the type, this reduces to a condition onthe cost function. With employer learning, it implies a condition on the relation between lifetimeincomes (as a function of beliefs) and costs (as a function of education). However, in a modelwhere education does not affect productivity, there is no direct relation between the former andthe latter, and so the condition can only hold if the set of possible value and/or cost functions isarbitrarily restricted.16

As an illustration, consider the value functions for high and low types depicted in Fig. 4below. Focus on beliefs relatively close to µ = 1. With employer learning, high-productivityworkers face a tradeoff between signaling through education and letting their performances on-the-job reveal their type. This is why the marginal return of an increase in the receiver’s beliefshould eventually decrease, which is not true for low-productivity workers. It is therefore naturalto expect that, for certain action-signal pairs, low-type workers prefer an education and beliefincrease while high type workers do not. This kind of effect is excluded by A4 because thiscondition prescribes that, whenever an increase in education is (weakly) attractive for the lowtypes, it must be (strictly) attractive for the high types.

Since the crucial condition A4 fails in our model, the analysis and uniqueness result in Choand Sobel [10] do not apply, and so there is a priori no reason to expect refinements like Uni-versal Divinity to generically restore equilibrium uniqueness in the absence of strong additionalrestrictions. Indeed, one can build (cumbersome) examples showing that, for general cost andvalue functions, those criteria do not necessarily refine the set of equilibria. In other words, anyattempt to further refine the equilibrium set using stronger refinement concepts must rely onadditional conditions. Daley and Green [12] investigate this issue by using the D1 criterion toanalyze a signaling game with grades that is similar to our static Reports Model. They proposea condition, which they label RC-Informativeness, under which correlated information is pre-cise enough relative to the cost advantage of high types. They show that D1 excludes all but oneequilibrium when grades are RC-Informative. Interestingly enough, the equilibrium outcome dif-fers from the Riley equilibrium as the latter fails D1 under RC-Informativeness. Hence, Daleyand Green’s analysis also illustrates that forward induction is vulnerable to the introduction ofcorrelated information after the signaling stage.

Our second observation is related to welfare concerns and the Pareto-efficiency of equilib-ria. Essentially, we will argue below that pooling equilibria in our model survive the IntuitiveCriterion only if the Riley separating equilibrium is Pareto-dominated. This delivers a connec-tion between our results and a different kind of equilibrium refinement criteria, built on a basicmessage of the signaling literature, namely that signaling might lead to inefficiencies. Mailathet al. [24] postulate the Undefeated Criterion as an alternative to the Intuitive Criterion and ar-gue in favor of pooling equilibria when they Pareto-dominate the Riley equilibrium. The keyargument is that, for some parameter constellations, the pooling equilibrium with zero educa-tion level might Pareto-dominate the Riley equilibrium: in the latter, high types overeducate to

16 Suppose we fix a value function and then require condition A4 to hold for any cost functions fulfilling the single-crossing property. Selecting the education levels in such a way that low types are indifferent and making the costdifferences arbitrarily close across types, a limit argument leads to v(ph,µ') & v(ph,µ) ! v(pl,µ

') & v(pl,µ) forall pairs µ' , µ. Hence, !v(ph,µ)/!µ! !v(pl,µ)/!µ. However, this contradicts P1: for µ " ]0,1[, v(ph,µ) > v(pl,µ)

and the inequality between the derivatives imply that ph/r = v(ph,1) > v(pl,1) = ph/r .


avoid low-type wages even though they would be better off with a pooling wage and minimaleducation.17

Suppose some pooling equilibrium survives the Intuitive Criterion. Then, as our analysisshows, the pooling equilibrium with e = 0 will also survive this criterion. By definition of theRiley equilibrium education level eh, the low type is indifferent between sending eh in orderto obtain the wage ph and receiving the wage pl at an education level of zero. The payoff atthe pooling equilibrium with e = 0 is strictly larger than the latter and so the Riley signal eh isalways equilibrium-dominated for the low type. If pooling at e = 0 survives the Intuitive Crite-rion, it follows that the payoff of the high types at the pooling equilibrium with e = 0 must beweakly better than at the Riley equilibrium. It follows that the pooling equilibrium with e = 0Pareto-dominates the Riley equilibrium. Repeating the argument for pooling equilibria with e

close to zero, it can be shown that, if any pooling equilibrium survives the Intuitive Criterion,the Riley equilibrium is defeated in the sense of Mailath et al. [24]. Observe, however, that anypooling equilibrium with e > 0 is defeated by the pooling equilibrium with e = 0. Hence, in ourframework, there will be defeated equilibria which survive the Intuitive Criterion.

A similar connection between welfare and equilibrium stability is identified in Daley andGreen [12]. They show that when correlated reports are RC-Informative, D1 always select equi-libria that Pareto-dominate the Riley equilibrium. These results suggest that learning restoresa link between signaling and efficiency. Recent work by Atkeson et al. [5] study a related issuein a model where firms may invest in product quality but cannot signal their choice. They findthat learning mitigates the ‘lemons problem’ as it introduces reputation incentives such that somefirms find it profitable to invest. They also show that welfare can be improved by the interventionof a regulator. Accordingly, it would be interesting to investigate whether existing labor marketregulations, such as income taxes and contractual obligations, can be welfare enhancing in ourset-up.

5. Bayesian learning

In this last section, the microfoundation of the learning process is discussed in more detail.We propose two models where beliefs are explicitly derived from output realizations, and showthat they deliver value functions satisfying the properties listed in Definition 2.1.

Note that output realizations are not decisions of the workers, but rather objective signalsobserved by the industry. Thus the microfoundations developed here concern employer learningbut do not correspond to further signaling stages on the side of the workers.18

17 The Undefeated Criterion is more involved than Pareto-dominance. Mailath et al. [24] restrict attention to pure-strategy signaling equilibria. Within this class, an equilibrium (" ',µ') is defeated if a new equilibrium (",µ) can bebuilt where a previously unused signal m is used by some types, in such a way that (i) the new equilibrium is a Pareto-improvement for those types (all of them being weakly better off, and at least one being strictly better off), and (ii) forsome type using m in (",µ), the receiver’s out-of-equilibrium belief in (" ',µ') that the sender is of that type on seeing m

can not be explained through conditioning on the set of sender types who do use m in (",µ), even allowing for thepossibility that indifferent types might have randomized. In the context of Spence’s model, for certain parameter valuesthe Riley equilibrium is defeated by pooling equilibria with education levels close to zero, while pooling at zero isundefeated. For other parameter values, the Riley equilibrium is undefeated.18 Models where signaling can occur over several periods give rise to a different game-theoretic structure. For example,Bar-Isaac [7] analyzes a signaling game where a monopolist sells a good whose quality is uncertain. Whether the actualquality of the product will be revealed over time or not becomes a question to be answered in equilibrium, rather thana postulated property of the learning process.


5.1. Two-period model

We first consider a set-up with two production periods, where output is observed by the in-dustry at the end of the first period. Realizations are not deterministic but randomly drawn froma continuous density gi(·) with a mean equal to the worker’s type pi for i " {l, h}. The sam-pling distributions share the same support and are common knowledge. The randomness mightbe inherent to the production process or to the imperfect precision of the monitoring technology.

Lifetime incomes follow, as in the two-period version of the Reports Model, from Eq. (2). Theupdating rule differs because second period beliefs now depend on the output density

µ'(x,µ) = µgh(x)

µgh(x) + (1 & µ)gl(x),

whereas wages w(µ) = pl + (ph & pl)µ are given by an affine function of beliefs.19

Proposition 9. The value function for the two-period model satisfies property P1 and so exhibitsweak learning.

A specific definition for s in this model would require further parametric restrictions onthe g(·) distributions. In any case, learning is weak because P3 does not hold for any con-ceivable measure of s. To see this, consider the limit case where uncertainty becomes negli-gible as gh(·) and gl(·) converge to Dirac delta functions. Then types are perfectly revealedat the end of the first period. Nevertheless, the lifetime income of high types converges to

11+r [w(µ) + v(ph,1)] < v(ph,1) = ph/r .20 As in the two-periods Reports Model, the upperlimit property in P3 is not fulfilled because first-period earnings solely depend on the prior andthus do not vary with the precision of correlated information.

5.2. Continuous-time model

In the two-period model, the length of time required to elicit information is a primitive pa-rameter. This is an artifice of discrete time, as industries where learning is more efficient shouldalso be characterized by shorter periods of information acquisition. This issue can be addressedusing a continuous time set-up and establishing that it gives rise to strong learning in the senseof Definition 2.1.21 The key difference, however, is not continuous vs. discrete time, but ratherthe fact that in the continuous-time model there are infinitely many future updates at any givenpoint in time, while in a two-period setting there is exactly one update and the production periodbefore that update never becomes negligible (see Anderson and Smith [3] for further detail onthis point).

We assume that output realizations are random draws from a Gaussian distribution with a timeinvariant average productivity.22 Thus the cumulative output Xt of a match of duration t witha worker of type i " {l, h}, follows a Brownian motion with drift

19 We do not report the proof of Proposition 9 in Appendix A because one simply has to use Jensen’s inequality in orderto show that E[µ'|pl,µ] < µ < E[µ'|ph,µ].20 Conversely, the lifetime income of low types converges to 1

1+r [w(µ) + v(pl,0)] > v(pl,0) = pl/r .21 We thank an anonymous referee for suggesting an alternative way to obtain strong learning in a continuous timeset-up: Let reports of the type described in paragraph IV arrive at the rate 1/" . Then strong learning would follow whenboth frequency " and accuracy s go to infinity.22 Letting workers accumulate general human capital would not substantially modify our conclusions.


dXt = pi dt + " dZt ,

where dZt is the increment of a standard Brownian motion. The cumulative output ,Xt - is ob-served by both parties. The employer uses the filtration {FX

t } generated by the output samplepath to revise his belief about pi . The variance " is constant across workers for otherwise firmswould be able to infer types with arbitrary precision by observing the quadratic variation of ,Xt -(see e.g. Chung and Williams [11]).23 Starting from a prior µ0, the employer applies Bayes’ ruleto update his belief µt " Pr(p = ph|FX

t ). His posterior is therefore given by

µ(Xt , t |µ0) = µ0gh(Xt , t)

µ0gh(Xt , t) + (1 & µ0)gl(Xt , t), (8)

where gi(Xt , t) " e& (Xt &pi t)

2

2"2 t is the rescaled density for a worker of type i, and the factor["

.2# t ]&1 is omitted because it simplifies in (8). The analysis is simplified by the change

of variable $t " µt/(1 & µt). $t is the ratio of “good” to “bad” belief. Given that µt is definedover ]0,1[, $t takes values over the positive real line. It follows from (8) that

$(Xt , t |$0) = $0gh(Xt , t)

gl(Xt , t)= $0e

s" (Xt& 1

2 (ph+pl)t), (9)

where s = (ph & pl)/" is the signal/noise ratio of output. On the one hand, a larger produc-tivity difference between types increases the informativeness of the observations. On the otherhand, a higher variance hinders the industry’s ability to identify the mean of the output distribu-tion. Thus the bigger s, the more efficient learning is. By Ito’s lemma, the stochastic differentialequation satisfied by the belief ratio reads

d$(Xt , t |$0) = $(Xt , t |$0)

%s

"

&[dXt & pl dt]. (10)

Replacing in (10) the law of motion of Xt , i.e. dXt = pi dt +" dZt , yields the following stochas-tic differential equations: (i) d$t = $t s dZt for low types; and (ii) d$t = $t s(s dt + dZt) for hightypes. The belief ratio $t increases with time for high types and follows a martingale for lowtypes.24 In both cases, a higher " lowers the volatility of beliefs because larger idiosyncraticshocks hamper signal extraction. We show in Appendix A how these two equations can be usedto derive closed-form solutions for expected lifetime incomes as a function of types and currentbeliefs.

Proposition 10. For the continuous-time model, the expected lifetime incomes of workers asa function of the belief ratio $ are given by

v(pl, $) = 2"

s%

+

$&&$,

0

1(1 + x)x&& dx + $&+

#,

$

1(1 + x)x&+ dx

-

+ pl

r

23 There are well-known issues with decision problems in continuous time. We refer the reader to Faingold [15] fora discussion of the significance of information revelation in the limit of discrete-time games, and to Alós-Ferrer andRitzberger [1] for a discussion of decision problems defined in continuous time. None of those issues apply to oursetting, because the continuous-time process captures information acquisition only and workers take no active decisions.24 It may be surprising that the belief ratio $t does not drift downward when the worker is of the low type. This isbecause the belief ratio $t is a convex function of µt . Reversing the change of variable shows that, as one might expect,the belief µt is a strict supermartingale when the worker’s ability is low.


Fig. 4. Workers’ value functions in the continuous-time model. Parameters: r = 0.2, pl = 0, ph = 1.

and

v(ph, $) = 2"

s%

+

$' &$,

0

1(1 + x)x' & dx + $' +

#,

$

1(1 + x)x' + dx

-

+ pl

r,

where &+ = 12 (1 + %), && = 1

2 (1 & %), ' + = 12 (&1 + %), ' & = 1

2 (&1 & %), and % =.1 + 8

"rs2

#.

The value function satisfies properties P1, P2, and P3, and hence exhibits strong learning.

The closed-form solution is of independent interest. The fact that it can be derived is anadditional contribution of this paper because continuous-time models often have to be solvednumerically. As discussed in Appendix A, the derivation crucially hinges on the change of vari-able from µ to $ . This technique can be used to simplify a variety of models with continuoustime learning, as illustrated by Atkeson et al. [5] or Daley and Green [13].

Plots of the value functions for a particular numerical example and several values of s areshown in Fig. 4, illustrating the properties listed in Definition 2.1. First, for any given beliefµ " (0,1), the expected incomes of low and high types are respectively smaller and bigger thanthe discounted value of their current wage, hence P1 is satisfied. Second, as stated in P2, thegap increases when learning becomes more efficient. Finally, the value functions converge to thediscounted value of current wages when s goes to zero and to step functions when s goes toinfinity, as required by P3.

6. Conclusion

We have analyzed a labor market where a worker’s ability can be revealed either by his edu-cation or by his performance on-the-job. The addition of this realistic feature causes the failureof standard arguments, such as the selection of the Riley equilibrium via the Intuitive Criterion.


Available evidence on the speed of employer learning suggests that this observation could behighly relevant for empirical research on job market signaling.

Our analysis has concentrated on characterizing the set of signaling equilibria and has reliedon the Intuitive Criterion as the basic refinement. From a game-theoretic standpoint, a very inter-esting avenue of research concerns the relevance of more sophisticated equilibrium refinements.As argued above, however, existing forward-looking equilibrium refinements might not be en-tirely suitable for signaling models with receiver learning, which raises a number of theoreticalquestions: In which special classes of games could a uniqueness result be restored? How canexisting refinements be further adapted to a context with learning on the receiver’s side?

Another avenue for future research concerns the learning process itself, whose characteristicscan vary across specific models. Our general model abstracts from prevalent features of labormarkets. For instance, it ignores the importance of match-specific uncertainty. This additionalsource of noise hampers signal extraction and is thus likely to enlarge the parameter space wherethe Intuitive Criterion bites. Another implicit premise is that wages are a function of currentbeliefs. This is no longer true if employers can commit to employment contracts. It would beinteresting to see whether commitment reinforces the informativeness of education signals.

Such extensions would provide a more realistic description of how signaling operates in labormarkets. Our basic findings, however, apply to any signaling environment beyond the particularjob-market model we have focused on. This suggests plenty of scope for further research on theinteractions between learning and signaling.

Appendix A. Proofs

Proof of Proposition 2. We first prove that the given education levels can be sustained as a pool-ing equilibrium by the particular beliefs µ(ep) = µ0 and µ(e) = 0 for all e (= ep . These beliefstogether with ep = 0 lead to pooling because any deviation yields a strictly larger cost anda strictly lower value (recall that the value function is strictly increasing in µ). Consider ep > 0:Under the stated beliefs, selecting e = 0 is always preferred by both types to selecting any othere /" {0, ep}, for both yield the same lifetime income but the cost of e = 0 is strictly smaller. Henceit suffices to check that neither of the two types has an incentive to deviate from ep to e = 0. Thisyields the conditions

c(pl, ep) & c(pl,0)$ v(pl,µ0) & pl

rand c(ph, ep) & c(ph,0)$ v(ph,µ0) & pl

r.

By the single-crossing property, c(ph, ep)&c(ph,0) < c(pl, ep)&c(pl,0). Weak learning, how-ever, implies that v(ph,µ0)! v(pl,µ0) and so the first condition implies the second one. Giventhat costs are strictly increasing in education, there exists a unique ep > 0 such that the firstinequality is fulfilled if and only if ep $ ep . By continuity, that education level is uniquely de-termined by the condition given in the statement. It is straightforward to show that an educationlevel can be sustained as a pooling equilibrium under some belief system if and only if it can besustained under the beliefs specified above. Further, by P1

v(pl,µ0) & pl $ w(µ0) & pl = µ0(ph & pl) < ph & pl,

which, recalling the definition of eh in Proposition 1, implies that ep < eh.Last, assume strong learning. To see that the set of pooling equilibria shrinks as s increases,

simply notice that v(pl,µ0|r, s) is strictly decreasing in s by P2, hence the conclusion fol-lows from the equation defining ep . The fact that lims%# ep = 0 under strong learning followsfrom P3, i.e. the requirement that lims%# v(pl,µ|r, s) = pl/r . !


Proof of Proposition 3. Suppose that there are at least two education levels in the support of thelow types’ equilibrium strategy, e and e' with e < e'. As both must be optimal, it follows that

v"pl,µ(e)

#& c(pl, e) = v

"pl,µ

"e'## & c

"pl, e

'#,which, since c(pl, e) < c(pl, e

'), implies v(pl,µ(e)) < v(pl,µ(e')). Hence, µ(e) < µ(e') (re-call that v is strictly increasing in µ), proving that µ needs to be strictly increasing over choseneducation levels. This is only possible if µ(e') > 0, hence in equilibrium e' must also be in thesupport of the high types’ strategy. It follows that there exist at most one education level chosenonly by low types, and it must be the lowest one in the support of their strategy. A symmet-ric argument holds for high types, thereby establishing (i). The fact that el = 0 follows as inProposition 1.

Suppose that also e is in the support of the high types’ strategy. Then, by the single-crossingproperty, we obtain

v"ph,µ

"e'## & v

"ph,µ(e)

#= c

"ph, e

'# & c(ph, e) < c"pl, e

'# & c(pl, e)

= v"pl,µ

"e'## & v

"pl,µ(e)

#.

The strict inequality leads to a contradiction in Spence’s model because v(ph,µ) = v(pl,µ).One can then conclude that low types randomize among at most two education levels, wherethe lower one is only chosen by them. Symmetrically, high types randomize among at most twoeducation levels, where the higher one is only chosen by them.

For other value functions, the last equality does not lead to a contradiction. The analysis ofthe Reports Model in the main text shows that equilibria where Ep is not a singleton are indeedpossible.

It remains to show that Ep ) [0, eh]. If ep " Ep , low-productivity workers must weakly pre-fer ep to obtaining wage pl/r while acquiring education e = 0, hence

c(pl, ep) & c(pl,0) $ v"pl,µ(ep)

#& pl

r$ 1

r(ph & pl)

(where the last inequality follows from property P0). Then, ep $ eh follows from definitionof eh. !

Proof of Proposition 4. Denote g " g(ph)/g(pl). Condition (6) holds if and only if

d > d(g) " 12

'1 +

(1 & g

1 + 3g

). (A.1)

Define Dµ " µd + (1 & µ)(1 & d) " [1 & d, d] and Kd " d & g(1 & d). Observe that d !12 > g(ph)

g(ph)+g(pl)= g

1+g implies that Kd > 0. Straightforward computations show that

r · h(µ) = Kdw(µ,G) + (1 & g & Kd)w(µ,B),

r · h'(µ) = d(1 & d)(ph & pl)

'Kd

D2µ

+ 1 & g & Kd

(1 & Dµ)2

),

r · h''(µ) = &2(2d & 1)d(1 & d)(ph & pl)

'Kd

D3µ

& 1 & g & Kd

(1 & Dµ)3

).

First, we notice that if d $ g(pl)g(ph)+g(pl)

= 11+g , then there exists no multiple pooling equilibrium.

This is because then 1&g &Kd = 1&d &gd ! 0 and it follows that h'(µ) > 0 for all µ " [0,1],implying that (5) cannot hold. Thus we can restrict attention to d > 1

1+g from now on.


Under this restriction, or equivalently 1 & g & Kd < 0, we obtain that h''(µ) < 0 for all µ "[0,1] from its expression above. Hence h is strictly concave, a fact we will rely on below.

The rest of the proof proceeds in three steps.

Step 1. Condition (A.1) holds if and only if

/µ1,µ2 with µ1 < µ2 and h(µ1) = h(µ2). (A.2)

To see this, note that by differentiability and concavity of h, (A.2) holds if and only if thereexists µ " ]0,1[ with h'(µ) = 0. If µ = 1/2, then D1/2 = 1 & D1/2 and from the expressionof h'(·), one sees that h'(1/2) > 0. Since h''(·) < 0, it follows that h'(·) has a zero in ]0,1[ if andonly if h'(1) < 0. Given that D1 = d , this holds if and only if d(1 & d) < g

1+3g .Observe that: (i) g/[1 + 3g] < 1/4; (ii) d(1 & d) reaches a maximum value of 1/4 at d = 1/2

and is strictly decreasing for d ! 1/2. Hence the condition is fulfilled if and only if d is above thelargest root of the polynomial of second degree above. That root is d(g), completing the proof ofthis step.

Step 2. If a multiple pooling equilibrium exists, (A.1) holds and part (a) of the statement fol-lows.

To see this, note that existence of a multiple pooling equilibrium implies (A.2), so (A.1) holdsby Step 1. Concavity of h implies that, for any given value k, there can be at most two beliefsµ1, µ2 with µ1 < µ2 and h(µ1) = h(µ2) = k. It follows that, if a multiple pooling equilibriumexists, it involves exactly two pooling signals, as claimed in (a), i.e. Ep = {e1, e2} with e1 < e2.

To complete the proof of (a), it remains only to show that there is no additional separatingsignal for high types. Suppose there were a third education level, e3, sent in equilibrium byhigh types only. High types must be indifferent between e3 and e2 and low types must weaklyprefer e2, hence

e2 + (ph/r) & v(ph,µ2)

g(ph)= e3 > e2 + (ph/r) & v(pl,µ2)

g(pl),

which holds if and only if h(µ2) < h(1). This leads to a contradiction, because concavity of hand (A.2) imply h'(µ) < 0 for all µ " [µ2,1].

Step 3. If condition (A.1) is fulfilled, then a multiple pooling equilibrium as in (b) exists. Further,there exists µmin " ]0,1[ such that a multiple pooling equilibrium as in (c) exists if and only ifµ0 > µmin.

By Step 1, condition (A.2) holds. Let µ1 < µ2 be such that h(µ1) = h(µ2). We aim to con-struct multiple pooling equilibria with Ep = {e1, e2}, µ(e1) = µ1 and µ(e2) = µ2.

For given {e1,µ1,µ2}, e2 is uniquely determined since v(pl,µ1) & g(pl)e1 = v(pl,µ2) &g(pl)e2. Consider e1. A sufficient condition for low types not to have an incentive to deviateunder some out-of-equilibrium belief system is the participation constraint or, equivalently,

e1 $v(pl,µ1) & (pl/r)

g(pl)"*e(µ1).

Note that *e(µ1) > 0 for all µ1 > 0. If low types send an additional separating signal e0 = 0 inequilibrium, indifference leads to e1 =*e(µ1). The analogous condition for the high types is re-dundant, since v(ph,µ1) > v(pl,µ1) > pl/r +g(pl)e1 ! pl/r +g(ph)e1 for all e1 " [0,*e(µ1)].Since this inequality is strict, e1 > 0, i.e. 0 /" Ep for any multiple pooling equilibrium. In sum-mary, e1 " ]0,*e(µ1)].


Let us turn to beliefs. Let q(e|p) denote the equilibrium probability of signal e sent by type p.By Bayes’ rule, weak consistency of beliefs implies, for j " {l, h},

µj = q(ej |ph)µ0

q(ej |ph)µ0 + q(ej |pl)(1 & µ0),

where µ0 is the share of high types in the population. To economize in notation, consider thechange of variable $ " µ/(1 & µ) for 0 < µ < 1 (the same change of variable considered inSection 5), and denote $k = µk/(1 & µk) for k = 0,1,2. We can then rearrange the last equationas follows

q(ej |pl) = q(ej |ph)$0

$j. (A.3)

Since by Step 1 high types randomize on Ep only, it follows that q(e1|ph) + q(e2|ph) = 1.Rearranging, it follows from Eq. (A.3) and q(e1|pl) + q(e2|pl) $ 1 that

q(e1|ph)

'1$1

& 1$2

)$

'1$0

& 1$2

). (A.4)

Hence, an equilibrium with multiple pooling requires $2 > $0, or equivalently µ2 > µ0 (elseq(e1|ph) > 0 would yield a contradiction).

We now show that an equilibrium of the type given in (b) always exists. Since h(0) =(1 & g)(pl/r) < (1 & g)(ph/r) = h(1), under (A.1) there always exist pairs (µ1,µ2) withh(µ1) = h(µ2) and µ2 arbitrarily close to 1. In other words, let µ be such that h'(µ) = 0 andlet ( " ]max{µ0,µ},1[. For any µ2 " ( , one can find a µ1 " ]0,µ[ such that h(µ1) = h(µ2).That is, choosing µ2 " ( , µ1 is uniquely determined by h(µ1) = h(µ2) and this guarantees thatboth types are indifferent on Ep by definition of h(·). Indifference with e0 = 0 for the low typesleads to e1 =*e(µ1) and the participation constraints are fulfilled. Since µ2 > µ0, our previousanalysis shows that q(e|ph) can be constructed and then q(e|pl) are given by Eq. (A.3). Thisproves existence of multiple pooling equilibria as stated in part (b).

We now turn to (c). For any e1 " ]0,*e(µ1)], the participation constraints are fulfilled asabove. For bipolar equilibria, the inequality in (A.4) binds. Then, for q(e1|ph) < 1 to hold, itmust be true that µ1 < µ0. If and only if this is satisfied, consistent q(e|ph) can be constructedwhile q(e|pl) are given by Eq. (A.3). Hence, it remains only to establish under which condi-tions do there exist pairs (µ1,µ2) with 0 < µ1 < µ0 < µ2 < 1 and h(µ1) = h(µ2). Given thath(1) = (1 & g)(ph/r) > (1 & g)(pl/r) = h(0), there exists µ with h'(µ) = 0, and h''(·) < 0.Thus such pairs exist if and only if µ0 > µmin where µmin is defined by 0 < µmin < 1 andh(µmin) = h(1). !

Proof of Proposition 6. We start with part (a). By continuity, there exists ) > 0 small enoughthat

c(pl, e) & c(pl, ep) <12

ph & pl

r0e " (ep, ep + )).

Let * = c(ph, ep + )) & c(ph, ep) > 0, so that c(ph, e) & c(ph, ep) > * for all e > ep + ). Fora value function with strong learning, lims%# v(pl,µ|r, s) = pl/r and lims%# v(ph,µ|r, s) =ph/r . Hence there exists s! such that, for all s ! s!,

ph

r& v(pl,µ|r, s) >

12

ph & pl

rand

ph

r& v(ph,µ|r, s) <

12

*.


It follows that, for s ! s!, (ED) fails, implying that e!(ep) > ep +), and (PD) fails, implying thate!!(ep) < ep + ). We conclude that e!!(ep) < e!(ep), or that the considered pooling equilibriumsurvives the Intuitive Criterion.

Part (b) follows from the fact that ) and * above are independent of ep for linear costs. !

Proof of Proposition 7. In this proof, we make the dependence of all involved quantities on sexplicit. Recall from Proposition 2 that pooling equilibria correspond to education levels in[0, ep(s)] and that, under strong learning, ep(s) is strictly decreasing in s and lims%# ep(s) = 0.

Recall also conditions (ED) and (PD). A pooling equilibrium with education level ep fulfillsthe Intuitive Criterion if and only if condition (PD) fails at e = e!(ep), where the latter educationlevel is defined by taking equality in condition (ED). We introduce

Ih(ep|s) " c(ph, ep) & c"ph, e

!(ep)#+ ph

r& v(ph,µ0|r, s).

By definition, the Intuitive Criterion fails at ep if and only if Ih(ep|s) > 0.

Step 1. Ih is strictly increasing in ep . Hence, for a given s, if the Intuitive Criterion fails at ep , italso fails at all larger education levels.

To see this, differentiate e!(ep) to obtain !e!(ep|s)/!ep = ce(pl, ep)/ce(pl, e!(ep)) > 0. Dif-

ferentiating Ih with respect to ep and applying log-submodularity yields

!Ih(ep|s)!ep

= ce(ph, ep)ce(pl, e!(ep)) & ce(ph, e

!(ep))ce(pl, ep)

ce(pl, e!(ep))> 0.

Step 2. There exists s!(0) such that all pooling equilibria fail the Intuitive Criterion for s < s!(0),and ep = 0 survives it for s ! s!(0).

As in the proof of Proposition 6, we can find s!(0) such that ep = 0 survives the IntuitiveCriterion for s ! s!(0) and fails it for s < s!(0). In the latter case, the conclusion follows fromStep 1.

Step 3. Let s ! s!(0). There exists a unique education level *e(s) such that Ih(*e(s)) = 0 andIh(ep) > 0 for all ep >*e(s). Further, *e(s) is strictly increasing in s and lims%#*e(s) = +#.

It follows from Step 1 that either Ih(ep) > 0 for all ep , and so all pooling equilibria fail theIntuitive Criterion, or there exists a unique education level *e(s) as stated. By Step 2, the formercase can only occur if s < s!(0). Notice that

!Ih(ep|s)!s

= &ce

"ph, e

!(ep|s)#· !e!(ep|s)

!s& !v(ph,µ0|r, s)

!s< 0,

where the inequality follows from P2 and the strictly decreasing profile of e!(ep|s) in s (seeequality in condition (ED)). Differentiating Ih(*e(s)) = 0 with respect to s now yields

!Ih(ep|s)!ep

!*e(s)!s

+ !Ih(ep|s)!s

= 0

which, since !Ih(ep |s)!ep

> 0 and !Ih(ep |s)!s < 0, implies !*e(s)

!s > 0.It follows that *e(s) is a strictly increasing function. Thus, if lims%#*e(s) = +# does not

hold as claimed, it must have an upper bound and so (by virtue of being increasing) a finitelimit L. Recall that e!(ep) > ep for all ep . As Ih(*e(s)|s) = 0, it follows from the definitionof Ih that lims%# c(ph, e

!(*s )) & c(ph,*e(s)) = 0, because lims%# v(ph,µ|r, s) = ph/r . Thisis a contradiction with e!(L) > L.


Step 4. For s ! s!(0), the rest of the proof follows.Recapitulating, ep yields a pooling equilibrium if and only if ep " [0, ep(s)], and in that case

survives the Intuitive Criterion if and only if ep " [0,*e(s)]. Since ep(s) is strictly decreasing andconverges to 0 as s % #, and *e(s) is strictly increasing and diverges to infinity (and *e(s!(0)) =0 < ep(s!(0))), they must intersect at a unique precision s such that part (b) holds below s andpart (c) holds above it.

It remains to consider the case where ce is log-linear. From the computations in Step 1 we seethat, in this case, !Ih/!ep = 0, while from Step 2 we see that !Ih/!s < 0. Hence, for a fixed s,either all pooling equilibria fail the Intuitive Criterion, or all survive it (the locus of *s becomesvertical). !

Proof of Proposition 8. A mixed-strategy equilibrium fails the Intuitive Criterion if there existsan e such that the two conditions in Proposition 8 are fulfilled. To see the only if part, note thatlow and high types are indifferent between signals in Ep , hence v(pi,µ(e)) & c(pi,µ(e)) isconstant for all e " Ep , given i " {l, h}. Hence, if equilibrium dominance for the low types andthe participation constraint of the high types hold for a given signal in Ep , they also hold for allsignals in Ep . !

Proof of Proposition 10. For a low ability worker, d$t = $t s dZt and thus the asset value solvesthe following Hamilton–Jacobi–Bellman equation

rv(pl, $) & 12($s)2v''(pl, $) = (ph & pl)

%$

1 + $

&+ pl,

since the wage is equal to the expected output w = E[p|$ ] = (ph & pl)($

1+$ ) + pl . We have tosolve a second order non-homogeneous ODE with non-constant coefficients. The homogeneousproblem satisfies an Euler equation25 whose solution reads vH (pl, $) = C1l$

&& +C2l$&+

, where&& and &+ are the negative and positive roots of the quadratic equation: &(& & 1) s2

2 & r = 0.Thus && = 1

2 (1 & %) and &+ = 12 (1 + %) with % = 1

s

.s2 + 8r . Notice that &+ & && = % and

&+ + && = 1.To solve for the non-homogeneous equation we use the method of variations of parameters.

The non-homogeneous term is composed of a non-linear function of $ plus a constant term. Thuswe can assume that the particular solution is of the form

vNH (pl, $) =!y1($)$&& + y2($)$&+$

+ pl

r.

Standard derivations yield the system of equations%

$&&$&+

&&$&&&1 &+$&+&1

&%y'

1($)

y'2($)

&=

%0

& 2"(1+$)$s

&.

Given that the Wronskian of the two linearly independent solutions is $&&&+$&+&1 &

$&+&&$&&&1 = &+ & && = %, the particular solution reads

vNH (pl, $) = 2"

s%

%$&&

,1

(1 + x)x&& dx + $&+,

1(1 + x)x&+ dx

&+ pl

r. (A.5)

25 Euler equations are second order homogeneous ODE of the form +v($) + &$v'2v''($) = 0, for given constants +

and &. They admit a closed-form solution as described in e.g. Polyanin and Zaitsev [27].


The bounds of integration and constants C1l and C2l of the homogeneous solution are pinneddown by the boundary conditions

v(pl, $)&&&%$%0

pl

rand v(pl, $)&&&%

$%#ph

r. (A.6)

Let us first consider the homogeneous solution. Given that $&& % # as $ 1 0, the first boundarycondition can be satisfied if and only if C1l equals zero. Similarly, because $&+ % # as $ 2 #,the second boundary condition allows us to set C2l equal to zero. All that remains is to determinethe integration bounds in Eq. (A.5). Consider the following function

v(pl, $) = 2"

s%

+

$&&$,

0

1(1 + x)x&& dx + $&+

#,

$

1(1 + x)x&+ dx

-

+ pl

r. (A.7)

Let us examine first the limit when $ 1 0. Given that $&& % # and/ $

0 [(1 + x)x&& ]&1 dx % 0as $ 1 0, we can apply l’Hôpital’s rule to determine the limit. Straightforward calculations showthat $&& / $

0 [(1 + x)x&& ]&1 dx % &$/[(1 + $)&&] % 0 as $ 1 0. A similar argument yields$&+ / #

$ [(1 + x)x&+]&1 dx % $/[(1 + $)&+] % 0 as $ 1 0.26 Hence, (A.7) satisfies the firstboundary condition in (A.6). Now, consider the limit when $ 2 #. We can again use l’Hôpi-tal’s rule because $&& % 0 and

/ $0 [(1 + x)x&& ]&1 dx % # as $ 2 #, so that $&& / $

0 [(1 +x)x&& ]&1 dx % &1/&& as $ 2 #. Similarly, we obtain $&+ / #

$ [(1 + x)x&+]&1 dx % 1/&+ as$ 2 #. Thus we have

lim$%#

v(pl, $) = 2"

s%

%1

&&& + 1&+

&+ pl

r= 2"

s

% &1&&&+

&+ pl

r= ph

r,

where the last equality follows from &&&+ = &2r/s2. We have established that (A.7) also satis-fies the second boundary condition in (A.6), which completes the derivation of v(pl, $).

The asset value for the high type can be derived following similar steps. We now show thatboth value functions exhibit strong learning. We establish each property in turn.

P1. This property is most easily established reversing the change of variable from $t to µt .

(i) Low ability worker: dµt = µt(1 & µt)s(&sµt dt + dZt ), (A.8)

(ii) High ability worker: dµt = µt(1 & µt)s"s(1 & µt) dt + dZt

#. (A.9)

By definition

v(pi,µt ) =+#,

t

e&r(,&t)E!w(µ, )|pi,µt

$d,, for all µt " (0,1) and i " {l, h}, (A.10)

For high ability workers, we know from (A.9) that µt has a positive deterministic trend:µt(1 & µt)

2s2. As w(µt) = µt(ph & pl) + pl is a linear function of µt , it follows thatE[w(µ, )|ph,µt ] > w(µt) for all , > t , and so v(ph,µt ) > w(µt )/r . Similarly, condition (A.8)shows that µt has a negative deterministic trend when the worker is of the low type, thus

26 Notice that/ #$ [(1 + x)x&+]&1 dx <

/ #$ x&&+&1 dx = $&&+

/&+ . Thus/ #$ [(1 + x)x&+]&1 dx is bounded for all

$ > 0 and the asset equation is well defined.


v(pl,µt ) < w(µt )/r . Finally, notice that when µt goes to zero or one, its stochastic compo-nent vanishes which provides us with the two boundary conditions.

P2. Differentiating (A.10) with respect to s yields

!v(pi,µt )

!s= (ph & pl)

+#,

t

e&r(,&t) !E[µ, |pi,µt ]!s

d,, for all µt " (0,1) and i " {l, h}.

It is therefore sufficient to prove that !E[µ, |pi,µt ]/!s is positive when pi = ph and negativewhen pi = pl . This follows from (A.8) and (A.9) as beliefs exhibit a negative trend for low typesand a positive trend for high types, both of them being increasing in absolute values with respectto s.

P3. The limit condition as s % 0 is satisfied because both deterministic and stochastic termsin (A.8) and (A.9) converge to 0. Accordingly, beliefs remain constant, i.e. lims%0 E[µ, |pi,

µt ] = µt for i " {l, h}. To establish the limit condition as s % #, let us focus first on hightypes. Notice that the belief ratio $t is a geometric Brownian motion and so $, (Z|ph, $t ) =$t exp( s2

2 (, & t) + sZ), where Z is normally distributed with mean 0 and variance " 2(, & t).

Given that µ($) = $/(1 + $), we have µ, (Z|ph,µt ) = [1 & µt + µt exp(& s2

2 (, & t) &sZ)]&1. Hence, for all ) > 0 and Z, there exists a signal/noise ratio s(),Z) such that1 & µ, (Z|ph,µt ) < ) for all s > s(),Z). It follows that lims%# E[µ, |ph,µt ] = 1 whichin turn implies that lims%# v(µ|ph) = ph/r . One can establish in a similar fashion thatlims%# E[µ, |pl,µt ] = 0 because µ, (Z|pl,µt ) = [1 & µt + µt exp( s2

2 (, & t) & sZ)]&1. !

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