a. michael spence - nobel lecture

8/9/2019 A. Michael Spence - Nobel Lecture

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SIGNALING IN RETROSPECT AND THE INFOR-MATIONAL STRUCTURE OF MARKETSPrize Lecture, December 8, 2001

by

A. M ICHAEL SPENCE1

Stanford Business School, Stanford University, 518 Memorial Way, Stanford,CA 94305-5015, USA.

INTRODUCTION

When I was a graduate studen t in econom ics at H arvard, I h ad th e p rivilegeof serving as rapporteur for a faculty seminar in the then-new Kennedy

School of Govern ment . Amon g other distinguished scholars, it included all of my thesis advisers, Kenneth Arrow, Thomas Schelling, and Richard Zeck-hauser. In the course of that seminar there were discussions of statistical dis-crimination and many other subjects that r elate to the incompleteness of in-formation in markets. One of my advisers came in one day with the strongsuggestion that I read a paper he had just read called The Market forLemons by George Akerlof. 2 I always did what my advisers told me to d o an dhence followed up immediately. It was quite e lectrifying. There we all founda wonderfully clear and plausible analysis of the performance characteristics

of a market with incomplete and asymmetrically located information. That,combined with my puzzlement about several aspects of the discussion of theconsequences of incomplete information in job markets, pretty muchlaunched me on a search for things that I came to call signals, that wouldcarry information persistently in equilibrium from sellers to buyers, or moregenerally from those with more to those with less information. 3 The issue, of course, was that signals are not terribly complicated things in games wherethe parties have th e same incen tives, i.e. where the re is a commonly un der -

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1

This paper is based on a lecture delivered by the author in Stockholm on the occasion of the100 th ann iversary of the found ing of th e Nobe l Prize. I would like to than k my fellow recipien tsof the Nobe l Prize in Econo mics this year, Professor George Akerlof and Professor Joseph Stiglitz,for th eir work an d t he ir inspiration, an d m y the sis advisers, Professors Ken ne th Arrow, Th omasSchelling and Richard Zeckhauser whose ideas and guidance got me launched on the study of market structure (particularly informational structure) and performance. My colleagues Profes-sors Edward Lazear and Mark Wolfson gave me a great deal of constructive input. I also owe agreat d ebt to Professors James Rosse and Bruce Owen, with whom I learne d an d tau ght in dustrialorganization and applied microeconomic theory at Stanford. It was a great group of youngpeople and a wonde rful time to be in th at part of the eld.2 George Akerlof, The Market for Lemon s: Qualitative Un certaint y and the Market Mechanism,Quarterly Journal of Economics .3 I believe is was Robert Jervis who introduced the terms indices and signals. Indices are attri-butes over which one has no control, like gender, race, etc. Think of them as unalterable attri-bute s of someth ing, not ne cessarily a pe rson. Signals are thin gs on e do es that ar e visible and thatare in part designed to communicate. In a sense they are alterable attributes. I thought it was auseful set o f distinctions an d te rmin ology and I still do.


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stood desire to communicate accurate information to each other. Even inthat case (sometimes called the pure coordination case) however, there arepotential problems of choosing among equilibria as illustrated in Schellingsbrilliant analysis of the u se o f focal poin ts and contextual information to solvecommun ication/ coordination problems when the p arties have been de-prived of the ability to communicate directly. 4 In markets where the issue is of-

ten undetectable or imperfectly detectable quality differentials, the align-ment of incen tives is typically imper fect and the incen tive of the high-qualityproduct owners to d istingu ish themselves and the incentive of the low-qualityowners to imitate th e signal so as to obscure the d istinction is fairly clear. Of course there is more to it, as one needs to know such things as who is in themarket per sisten tly and hence who has an incentive to establish a repu tationthr ough repeated p lays of the game. I am going to devote a good portion of this lecture to these issues, in a sense to revisit signaling, and then turn tosome other aspects of the informational structure o f markets that are r aised

by the parameter shifts caused by the proliferation of the intern et as a com-munication med ium in th e p ast few years. 5

I was asked recently by a somewhat incredulous questioner ( actually a jour-nalist) whether it was true that you could be awarded the Nobel Prize inEconomics for simply no ticing that there are markets in which certain parti-cipan ts don t know certain th ings that others in the market d o know. Ithought it was pr etty funny. It was as if th is had somehow been a closely guard-ed secret up un til about 1970, at least in economics. I clearly cann ot speak forthose who make the decisions about the Nobel Prize, but I suspect that the

corr ect answer to that question is no. What did b lossom at that time was a se-rious attempt by many talented economists to capture in applied microeco-nomic theory a whole variety of aspects of market structure and per forman ce.That work p roduced a par tial melding of theor y, industrial organization, la-bor economics, nance and other elds. An important early part of that ef-fort was the attempt to capture informational aspects of market structure tostudy the ways in which markets adapt, and the consequences of informa-tional gaps for market per formance.

Therefore, in answer to the question, we noticed that th ere are man y mar-

kets with informational gaps. These include most consumer durables, virtual-ly all job markets, many nancial markets, markets for various types of foodand pharmaceuticals, and many more. These informational gaps were widelyacknowledged an d those of us who taught app lied microeconomic theory,freely admitted th at these gaps might chan ge some of the per forman ce cha-racteristics, not to mention the institutional structure, of markets in which

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4 Th omas Schelling, The Strategy of Con ict .5 In econom ics and othe r social sciences, mod els abstract from reality and focus on structural fea-ture s of organizations or markets that drive outcom es. The re are e mbed ded and usually un statedparame ters that are n ot the focus of attention, as they do n ot nor mally change. When I refer to pa-rame ter shifts caused by the in tern et, I mean ch anges in structural eleme nts that former ly did n otchange. For example, most markets have geographic dimensions and boundaries that are impli-citly un derstood an d are not the focus of attention. The intern et h owever, has moved th ose bou nd -aries by collapsing time an d distance in the inform ation/ commu nication dimen sions of markets.


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they appear. But I think it is fair to say that we did not have much systematicknowledge based on theory of what those changes might be. And so wethought that applied microeconomic theory deserved an attempt to buildthese informational characteristics into models that captur e th e structure an dper formance of these markets with reasonably accurate assumptions about theex ante informational conditions. It was a very exciting time for all of us who

were involved. One of the wonderful aspects of receiving this award is that ithas triggered in me, and I think others, fond memories of the sense of disco-very and excitemen t. And in that context, I would like to take this opportu ni-ty to express my admiration and deep gratitud e to the man y extant youn g col-leagues with whom I worked and shared these ideas. They should, andcertainly in my mind, do share in the recognition tr iggered by awarding of thedistinguished Nobel Prize for what was accomplished during those years.

The p lan for th is lecture is as follows. The overr iding goal is not to look toostup id to the next generation of stud en ts and scholars. More seriously, fol-

lowing the advice that my advisers once gave me, I will rst discuss thesimplest model that I can devise that illustrates in reasonably general formthe den itions and p roperties of signaling equ ilibria. 6 Next, we will allow th esignal (in this case education in the job market context), to contributedirectly to the p roductivity of the ind ividual as well as fun ctioning as a signal.Following that, the paper examines a market in which there is signaling andboth separating and p ooling in the equilibrium.

The section after that examines a fairly general partial equilibrium model of signaling and discusses competitive equilibria and certain kinds of optimal re-

sponses to signals.7

Those who are more interested in th e general idea and lessin the general case can skip over this section without risk of missing any impor-

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6 Joh n von Neum ann , who d eserves to be on a short list of the great est minds of the 20 th cen tur y,is repo rted to h ave said th at you d ont und erstand a theo ry or an abstract structu re u ntil you h aveseen and worked through hundreds of examples of it. Even if he didnt say that (and I think hedid), I agree with it. Few people can say that they had the idea that calculating machinesdidn t have to be har d-wired, that you could store instructions in me mor y and the n e xecute th emin order: that is, you could build a programmable computer.7 These optimal responses problems are selection problems. In insurance markets, they can bemoral hazard or adverse selection problems, and in other contexts they are agency problems or

optimal taxation p roblems. You want to con front p eople with sched ules that cause th em to m akeappr opriate ch oices and in so doing to r eveal them selves and th e information that they privatelyhold ex ante. Generally the results are below what is achievable with perfect information, becausewith p er fect information , you d ont have to worr y about the revelation pr oblem. Th ese problem shave th e econ omic and math ematical prope rties of second -best taxation problem s, of which p er-haps the rst to be an alyzed was the optim al incom e tax pr oblem , by th e Revere nd Ramsey,th ough the an alysis seem ed to have largely escaped no tice un til Jim Mirrlees, Ton y Atkinson, an dthe n o ther s rediscovered it. The idea is that th e taxing auth ority canno t directly observe an indi-viduals earning power (think of it as maximum potential hourly wage) so they tax what they cansee and mon itor, which is incom e. Th is produces a d isincentive with r espect to work that couldhave been avoided if the tax were based on earn ing poten tial rather th an actual earn ings. The se-cond best optimizing calculation then trades off in the most efcient way disincentives for work and t he n eed to m eet a reven ue goal for the t axing authority. The math ematics of these problemsare sometime s referred to as optimal control p roblems and somet imes the calculus of variations.In later work, Holmstrm applies these models to a labor setting, incorporating Wilsons work onsyndicates (optimal risk-sharing). Here the second-best optimizing calculation sacrices optimalrisk- sharin g em ployees an d owne rs to p rovide incen tives for em ployees to work d iligen tly.


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tant properties. However, this is the section in which the formal relationship be-tween signaling models and models of self-selection and optimal taxation withimper fect information is examined , and that may be of interest to some readers.I also want to in troduce in that context a new possibility that I discovered only re-cently and that is that one can have a signaling equilibrium in which the costs of the signal appear to vary with the unseen ability characteristics in th e wrong way.

That is to say the costs of education ( absolutely and at the margin) rise with abi-lity, or more gen erally, with the unobserved attribute that contr ibutes positivelyto p roductivity. For the most part, I will use the job market case for expositionalpurposes and comment on some other signaling situations more briey later inthe essay. There is a risk in using job markets to illustrate signaling. In order toillustrate the fundamental properties of signaling models, I have stripped awayother features of the market, particularly in the simpler models. But I have no-ticed in the past that there is a tendency for the simpler models set in the jobmarket context, to convey unintended messages such as (1) education does not

contribute to productivity, or ( 2) the information contained in the signal doesnot increase efciency. This essay is main ly about the theory of signaling and in-formation transfer in markets, and not mainly about h uman capital and labormarkets. There are many who are more knowledgeable about the latter than I.

The paper concludes with two sections. The rst of these discusses theubiquitous use of time and the allocation of time as a signal and as a screen-ing device. The second focuses on the potentially rath er large information pa-rameter shifts that the internet may have caused and speculates a little onhow applied models of markets, organ izations, and bound aries between them

may have chan ged and on what is needed to captu re th e effects of these pa-rame ter shifts in models of markets and the economy.

TH E SIMPLEST JOB MARKET SIGNALING MODEL

The idea beh ind th e job market signaling model is that is the re are attr ibutesof poten tial emp loyees that the employer cannot observe and th at affect theindividuals subsequent productivity and hence value to the employer on the

job. Let us suppose that th ere are just two groups of people. Group 1 has pro-

du ctivity or value to any employer of 1, and Grou p 2 has productivity of 2. Inthis example, these productivity values do not depend on the level of invest-ment in the signal. If there were no way to distinguish between people inthese two groups then if both groups stay in the market, the average wagewould be 2 , where is the fraction of the population in group 1, andeveryone would get that wage. If the higher productivity group through dis-satisfaction or for any other reason, exits this labor market, the average pro-ductivity and the wage drop to 1. This phen omenon when it occurs is some-times called th e adverse selection p rob lem, a label most common ly app lied toinsuran ce markets. It is stru cturally the same problem that Akerlof describedin his famous paper on used cars ( lemons) . 8

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8 George Akerlof, The Market for Lemon s: Qu alitative Un certainty and t he Market Mechanism,Quarterly Journal of Economics , 1971.


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Now suppose that there is something called education, which we will de-note by E, that can be acqu ired or invested in . It is assumed to be visible, andits acquisition costs differ for the two types. Let us suppose th at th e cost of Eyears of education for group 1 is E, and the cost for group 2 individuals isE/ 2. For th is example I am going to assume that education does not affectthe individuals prod uctivity. I do th is pu rely to keep it simple an d n ot to sug-

gest that h uman capital, includ ing that acquired through edu cation , is some-how not relevant. In later sections, the assumption will be re laxed.

Equilibrium in a situation like this, and in general, has two components.First, given the re turns to and th e costs of investing in education , individualsmake rational investmen t choices with re spect to education. Second, employ-ers have beliefs abou t the r elation be tween the signal and th e ind ividuals un -derlying produ ctivity. These beliefs are based on incoming data from the mar-ketplace. In equilibrium, the beliefs must be consistent, that is, they must not be disconrmed by the incoming data and the subsequent experience. There-

fore, one could say that the beliefs must be accurate . But on e should also no-tice th at employers beliefs/ expectations determ ine the wage offers that aremade at various levels of education. These wage offers in turn determine theretur ns to ind ividuals from investmen ts in education , and nally, those re-turns de termine the investmen t decisions that individuals make with r espectto education, and hence the actual relationship between productivity andeducation that is observed by employers in the marketplace. This is a com-plete circle. Therefore it is probably more accurate to a say that in equili-br ium, th e employers beliefs are self-conrming . This may sound like a minor

restatemen t bu t it is impor tant. It is the self-conrming n ature of the beliefsthat gives rise to the potential presence of multiple equilibria in the market.

In th e example at h and , suppose that group 1 individuals set E 1 = 0 and in-dividuals in group 2 set E 2 = E*. Suppose further that employers, none of whom individually inuence investment decisions by individuals 9, believe thatif E < E* then product ivity is 1 and if E E *, then productivity is 2. With theseassumption s, ind ividuals in group 1 will ration ally set E = 0 provided that

2 E * < 1

Members of group 2 will ration ally set E = E* provided that

Therefore th e choices will be rational and th e expectations conrmed in themarket if

1 < E* < 2.

Though it is a highly stylized example it has many of the general propert ies of

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9 Investment decisions are made by individuals in advance of knowing with whom they will work and emp loyers are such th at no ne of them individually inu en ces materially the per ceived offersin the m arket place.


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signaling equilibria. There is a continuum of equilibria, in each of whichthere is more investmen t in th e signal than there would be in a world of fullinformation. Because investment in the signal dissipates resources withoutimproving productivity, the result is inefcient. Moreover the equilibria areorderable by the Pareto criterion , that is, as you move from one equilibriumto another, everyone is either worse off or no better off. The signal actually

does distinguish low- and h igh-productivity people and th e reason it is able todo so is that the cost of the signal is negatively correlated with the unseencharacter istic that is valuab le to em ployers, in th is case productivity itself. Theequ ilibrium is characterized by a sched ule that gives the re turns to education ,that is a wage for each level of education, and optimizing choices given thatschedu le by ind ividuals in the two group s.

The equ ilibrium is depicted in Figure 1. The wage schedu le is the dar k lineand jumps from a wage level of 1 to a wage level of 2 at e*. E* is between 1and 2 so that group 1 chooses the edu cation level 0 and group 2 chooses E*.

It is not essential that net income for group 1 at e* is negative as it is in thisexample. We could make the productivity levels for the two groups 3 and 4.The same signaling result would occur with suitable adjustments in the levelof the wage schedu le.

The re are in fact other signaling equilibria. For example, there could be aminimal level of education that group 1 invests in. But unless it is a produc-tive investmen t, ther e is no reason to th ink that over time the market wouldnot discover th is and eliminate it as an inefciency that can be r emoved cost-lessly. In th e spectru m of equ ilibria described above, probably the most in te-resting is the most efcient one. That equilibrium is the one in which

E * =1 + , where is to be thought o f as a small positive number. In this equi-librium,

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Figure 1


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w1 = 1 = 1 , an d

w2 = 2, and N 2 = 1.5 / 2,

where N i is income net of signaling costs for group i.On the assumption that th e market will nd equilibria that are Pareto-ef-

cient, one can ask whether there is a better equilibrium in the Pareto sense(everyone is better off) than the one described above. The answer is some-times yes. It involves pooling and it depends on the relative sizes of the twogroups. What is really going on here is that the market is, in a sense, acting asif it is trying to maximize the net incom e of the h igher productivity group. Insome cases that is accomplished by recognizing that it is too expensive forthat group to d istinguish itself. The altern ative is to appear u nd ifferentiatedin a pool. That clearly benets group 1 as they are mistaken for the averagepr oductivity, which is above 1. As a reminder, is the fraction of the popula-

tion in group 1. In a pooling situation the average productivity is 2 > 1.Group 1 would clearly pr efer it. Group 2 prefers it as well if

2 > 1.5 / 2, or

< .5(1 )

Since can be mad e as small as we want, the re is a pr eferred pooling equ ilib-rium (in the Pareto sense) if in this example, group 1 is less than half thepopulation. In general with discrete groups, pooling with lower level groups

becom es attractive if they are relatively small, because th e h igher level groupsdont give up too much by being mistaken for the average and they avoid thesignaling costs. In a later section I will describe br iey a case in which on e canobserve pooling and separating in th e same equ ilibrium.

If you wanted to make the result in this market more efcient than theequilibria described above, you would tax education (assuming that such atax is not costly to impose and administer), making it more expensive forgroup 2, and thereby make it possible to lower th e level of education withoutlosing the informational conten t of the signal. The revenues generated could

be d istributed equally to all participants in both groups indep enden t of theiredu cation choice. The lump-sum component of the tax will therefore not af-fect their signaling beh avior.

Let t be the tax rate on investmen t in edu cation and let k be the lump -sumdistribution from th e tax revenues. This distribution goes to everyone. Let thesignaling level of education be E*. Group one individuals will rationallychoose no t to send the signal if

2 (1 + t ) E * + k < 1 + k

and group 2 will rationally sen d the signal if

2 (.5 + t ) E * + k > 1 + k .

We h ave an equ ilibrium th en if

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We will pick the efcien t en d of the signling spectrum by setting

The lump sum distribution is equal to the tax revenues so that

Thu s the equilibrium n et incomes are

As becomes small and t becomes very large, k approaches (1 ), andhen ce both net incomes approach ( 2 ). This, the reader will recall, is thepooling equilibrium outcome in term s of net income. Here however, the re isno upper limit on th e size of group 1, namely . Signaling costs (socially) arenegligible as E* app roaches zero, though th e private marginal costs of signal-ing (including the tax) approach 1.

To summarize, one can get rid of the inefciency and keep the signalingand the informational content of the signals with an appropriate tax on thesignaling activity. The effect is to redistribute income in such a way as to re-produce the results of the p ooling equ ilibrium. It could be that th e informa-tion the signal carries is itself productive. In that case it is important that theequilibrium actually be a separating one. 10 That is the case h ere. Separating isretained in this equilibrium. Group s are cor rectly identied. Th e use of a taxon the signaling activity reduces the level of the signal requ ired to d istinguishthe groups, and h ence redu ces the inefciency of the un taxed signaling equi-librium.

TH E TWO-GROUP MODEL WHEN EDUCATIO N ENHANCES HUMANCAPITAL AND ALSO SERVES AS A POTENTIAL SIGNAL

It will natu rally occur to th e re ader to ask what is the effect on the market sig-naling that we have just examined, at least in the context of labor markets,when the signal also contributes directly to productivity for the individualworker. 11 This section addresses the subject by modifying the two-group mo-del of the preceding section to allow education to be productive. I will make

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10 Inform ation is directly prod uctive, for examp le, if the re is a job allocation o r train ing de cisionthat is mad e mo re e fciently with accurate knowledge of the type. In that case th e po oling equili-brium would be inefcient.11 Actually th is issue does not just arise in labor market s. Gen era lly signals can change th e value o f the produ ct. For example, in th e used car case, a warrantee changes the value of the bu ndle, thecar and the warrantee in addition to carrying information about th e pr oduct itself.

an d

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the model slightly more gen eral in te rms of fun ctional forms. Let s i(E) be thevalue of a worker of type I with education E to an employer. Here I = 1,2 andwe assume that s2( E ) > s1( E ), and s2 ( E ) > s1 ( E ). Let c i(E) be the cost of in-vesting in E units of education for group I and assume th at c1( e) > c2( e), andthat c1 ( E ) > c2 ( E ) . These assump tions simply capture the idea that the groupwith the h igher productivity also h as the lower costs of signaling. As before,

the individuals type is not directly observable. We will also assume that Si( E )is concave, that C i( E ) is convex and hence that the n et income function N i( E )= Si( E ) C i( E) is concave.

There are three qualitatively different kinds of equilibria in this market.The r st is a fully efcien t separating equilibrium. It is easiest to see the typesof equilibrium pictorially. In Figure 2, we have various forms of net incomeplotted as a fun ction of E. Let V 1( E ) = S2( E ) C 1( E ). Our interest in th e func-tion V clearly re lates to th e question of wheth er group 1 will adopt the grou p2 signal and hence create a p roblem with the separating equilibrium.

In Figure 2, you will see the fun ctions N 1, N 2, and V. The points E 1* and E 2*are th e poin ts that maximize N 1 and N 2. The point E

is the largest value of E

such that V is larger th an N 1*, the maximum value of N 1. In Figure 1, you cansee that E 2* is to the right of E

. This means that as long as the wage offer

schedule jumps up from s 1(E) to s 2(E) to the right of E

, individuals in group1 will have n o incen tive to imitate th e signaling behavior of group 2.

This is the fully efcien t separat ing equilibrium . The signal carr ies accurate

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Figure 2


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information, and the investments in education are the efcient ones. Theoutcome is as if there was perfect information in the market place. Inessence, the two groups are sufciently different in terms of some combina-tion of the p roductivity of education as human capital and the costs of edu-cation that the fully efcient outcome survives as an equilibrium. Note how-ever, that the n egative relation between productivity and costs really matters.

If signaling costs were the same, then the functions N 2 and V would be thesame and we could not have this market outcome. When signaling costs arethe same one can never prevent group 1 from imitating group 2 behaviorwhen it is to th eir advantage to do so.

Now turn to Figure 3. It is essentially the same picture as Figure 2 exceptthat E 2* is to th e left of E

. This means that if the wage sched ule jumps to s 2(E)

before E 2*, the group 1 will imitate grou p 2 behavior and disman tle the se-parating equilibrium. To prevent this we will specify that the wage schedule

jumps up to s 2(E) at E

+ . Subject to a qualication that I will come to in a

minu te, th is is a separating equ ilibrium. To ach ieve th e separation, th e grou p2 investmen t in education is pushed u p above th e efcien t level, in order forthe individuals in that group to achieve the signaling effect and avoid imita-tion by group 1. So h ere we h ave th e same kind of result that we h ad in theno-human-capital case. Here the return to investing in education has a sig-naling component and a human capital component. The former can pushthe levels of investmen t above th e full information op timum. In the rst mo-del the h uman-capital effect was zero, leaving only the signaling effect, an as-

416

Figure 3


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sumption that ensures that overinvestment will occur in any separating equi-libr ium in which signaling occurs.

There remain two issues. One is whether and under what circumstances,pooling will destroy the hypothesized separating equilibrium and the secondis, Can one tax the signal in such a way as to improve market p er formance?Pooling rst. Let be the fraction of the total population in group 1. In a

pooling equilibrium, the wage for all will be s1( E ) + (1 ) s2( E ) for any le-vel of E that emerges. This cannot break a fully efcient separating equili-brium because group 2 cannot be better off. Thus if pooling breaks a sepa-rating equilibrium it can only be in the case where the more productivegroup has been forced, for signaling purposes, above its optimal level of Efrom the stand point o f investmen t in human capital. Let W ( E ) = s1( E ) + ( 1 ) s2( E ). By making arbitrarily small we can make W as close to s 2(E) as wewant. Now let us look again at Figure 2. If you start at E

and reduce E and

move to th e left and up th e cur ves N 2 and V, it is clear that both groups are

bet ter off. For small , the net income fun ctions W-c 1 and W-c 2 are close to Vand N 2. Hen ce for a ran ge of at the low end, pooling will break the sepa-rating outcome. For these cases, the natur al choices of equilibrium levels of Eare those that lie between the education levels that maximize W-c 1 and W-c 2.

As rises, eventually the function W-c 2 will fall below the level of group 2net income at the separating point E

, at which point the pooling ou tcome is

not capable of breaking the separating equilibrium. To summarize, in thecase of an inefcient separating equilibrium, there exist pooling equilibriathat are Pareto-superior to the separating equilibrium provided th e size of the

lower productivity group is below some threshold level. If the market has away of nd ing the pooling equ ilibria, they will break the separating on e.

We tur n n ow to the problem of impr oving the efciency of the market. Wesaw in the pr eceding section that in the case whe re th ere is no human capitalcomponent of productivity, one can in p rincipal come arbitrarily close to theefcient ou tcome th rough app rop riate taxes. The same is true when th e sig-nal adds to the individuals productivity as well as functioning as a signal. Wehave already seen that in some cases the market equilibrium itself is efcientin th e sense of maximizing total net income as part of a separating equ ilibri-

um. We now want to show that one can achieve the same result in the casewhere the higher productivity group has been forced by signaling consider a-tions to move above its efcient level of investmen t in h uman capital.

To achieve an efcient outcom e, we want to maximize total net income. Todo th at we need to have si = ci for i = 1,2. To accomp lish that if it is possible,we need a wage function that induces these choices. The wage function canbe thought of as a schedule of taxes on education superimposed on a wagefunction that sets wages equal to productivity. Since each individual in eachgroup considers all possible levels of E in m aking its education investmen t de-cision, it turns out to be easier to solve th is problem indirectly rather than di-rectly. Let us suppose therefore th at coun ter to fact, there is a continu um of types specied by the p arameter z, where productivity for type z is zs 1+(1-z)s 2and costs for type z are zc 1+(1-z)c 2. Let E(z) maximize

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z( s1 c1)+ (1 z) ( s2 c 2) .

Note that E(z) declines as a function of z. Inver t E(z) to get Z(E). If the wageschedu le we ar e looking for is w(e) , then ind ividuals will have m aximized bysetting

w = zc1 + (1 z) c2 .

Substitute Z(E) into the differential equation above and integrate with re-spect to E, setting the constant so that total wages equal total output. Thiswage schedule and / or th e implied tax schedule will ind uce the efcientchoices of E. 12

The reader n otices that th e optimal wage schedule above does no t dependon the distribution of z in the population. 13 Therefore let us assume th at al-most all the weight in the d istribu tion is at either z = 0 or z = 1. This is the twogroup case we are studying.

To summarize, with two grou ps and education as produ ctive h uman capi-tal, one can have a signaling equilibrium with full efciency or overinvest-ment in education by the more p roductive group. You can also have a poolingequilibrium that dominates the separating equilibrium provided th at the less-productive grou p is no t too large. And nally the re exists a tax/ subsidyscheme th at produ ces a fully efcient separating ou tcome as an equilibrium.

A MODEL WITH SIGNALING, SELECTION AND PO OLING 14

The model we are going to look at br iey in th is section is of inter est for tworeasons. First it illustrates a case in which there is both pooling an d separ atingcomponents of the equilibrium, and second , it shows that the critical criteri-on for having a separating component in an equ ilibrium is that the ne t bene-ts of issuing the signal are positively correlated with an un seen attr ibute thatcontributes positively to productivity. This positive correlation can result as inpreceding examples, from signaling costs that are negatively correlated withthe valued attribute. That is a sufcient but not a necessary condition for aseparating equilibrium. In this very nice example the information is acquiredat a xed cost an d the positive cor relation comes from subsequen t d iscoveryof the attribute post-emp loymen t.

The idea behind th e model is that the value of individuals to rm s is not di-rectly observed , at least at the time of hiring. The value which we will den oteby q is distributed on the interval [q min ,q max ] . Let the distribution of q in the

418

12 Note that w [ zc1 + (1 z) c2] = z ( E ) [ c1 c2 ) < 0, so that the second or der con dition for amaximu m is satised. Th e op timal tax schedu le will actually consist of a r ising tax on edu cationat all levels of education combined with a lump sum distribution to everyone so that net tax re-venu es are zero. In this example, grou p 1 will receive a ne t subsidy and grou p 2 will pay a n et tax.13 This is a gen eral p rope rty of the contin uou s case, whe n the objective is, as it is he re, to maxi-mize net incom e by indu cing the e fcient levels of investme nt in h um an capital.14 This example was developed by Professor Edward Lazear. It illustrates nicely the coexistence of pooling and separating in equilibrium and also a differen t structural condition that perm its thesignaling an d selection to occur. It can be foun d in his book, Edward Lazear, Personnel Economics .


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pop ulation be f(q) . Individuals have a choice. They can work for rms that donot distinguish among them and hence pay everyone the same amount. Orthey can work for rms that incur an expense, denoted by e, as a result of which, the rm even tually learn s the value of q for the ind ividual and pays theperson accord ingly. In equ ilibrium th is cost, which is the same for everyone,is passed on to the individual in the form of a reduction in compensation.

Labor markets are assumed to be competitive. For those individuals whochoose the rms where they are distinguished, the wage is q-e. If the indi-vidual chooses a rm that does not incur the cost and does not distinguishamong worker s, then the comp ensation is the average value of the individualswho work for that rm.

Let us suppose that the average value of the workers in the pooling rms isq . If we con sider the optimizing decisions of ind ividuals, it is clear th at if

q e > q

then the individual will choose to work for the separating rm and converse-ly. Let q* = q + e. In equilibrium, q is the actual average value of those whowork in the pooling rms, so th at

where F(q) is the cumulative distribution function for the attribute q. The

equ ilibrium is illustrated in Figure 4.

419

Figure 4


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Those with q q* receive q and ar e pooled , while th ose with q > q* receiveq - e and are iden tied and ar e in the separating part of the equ ilibrium. It ispossible that everyone will be pooled. This occurs when e, the cost of disco-very, is large en ough that q max e is less than the un cond itional mean of thewhole distribution . Another view of the equ ilibrium is provided in Figure 5.

Here we plot the functions q*, and E ( q | q q* ) + e, as a function of q*. The

secon d function is upward -sloping and will cross the 45-degree line, unless eis large enou gh to keep th e second fun ction above th e 45-degree line for allq*. The crossover point is the equ ilibrium level of q*. 15

The signal is the choice of which type of rm to go to and it is par tially, butnot complete ly, informative about th e ind ividuals value to th e rm. I notedin the rst model that th e signal retains its informational conten t in equilib-rium if the cost of the signal is negatively correlated with the (hard-to-ob-serve) valued attr ibute . In th is case, the cost of the signal is a constan t acrosspeople but because of the subsequent discovery and adjustmen t, the n et be-

net to the individual of issuing the signal is positively correlated with thevalued attribute.

It is possible that th e information carr ied by the signal increases efciency.That circumstance increases the return to going into the separating side of

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15 While th e con dition al expected value fun ction is upward-slop ing, its slope is no t necessarily lessthan one, and depen ds on f(q). It is therefore in p rincipal possible to h ave an o dd n umber of multiple crossovers. If there are three, the middle one will be unstable and the ones at the out-side will cross over with a slope of less than one and be stable. To see this, note that if at somepoint of time, q* e > q, th en q* will rise, an d co nversely.

Figure 5


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the market an d h as the overall effect of increasing the size of the group th atsends the signal. I should also note that it has been implicitly assumed herethat the inform ation about productivity, once it is acquired, is public. Thatwould force the emp loyer to pay the emp loyee h is or her p roductivity minusthe d iscovery cost. If, on the other hand, th e d iscovery is pr ivate, th ere will bea negotiation between the employer and the employee over the net income,

and that will have th e effect of lowering th e n et income of the employee r e-lative to the case in which th e information is public. Thus one should expectthe size of the pooling compon ent of the market equilibrium to rise when thediscovery is private.

THE GENERAL CONTINUO US MODEL 16

I would now like to establish some reasonably general properties of signalingmodels. 17 Following that we will look at improving the equilibrium perfor-

man ce of the m arket through taxes and subsidies, using the app roach of op-timal taxation with imper fect information. Th e variables are n, stand ing forsome attribute that is (a) not directly observable, and (b) valuable to em-ployers, and y, which is years of education . The latter is observable an d maybe valuable to emp loyers. The functions with n eed are S(n,y) which speciesan individuals productivity or value to an employer as a function of educa-tion an d n , which I will hen ce forth refer to as ability. The r emaining fun c-tions are c( y,n) which species the cost of education as a function of the sametwo variables and w(y) which is the wage offered to an individual who pre-

sen ts him or herself in the market with education of y. Since th is is quite fa-miliar terr itory, I will proceed fairly quickly. The equilibr ium is dened by twoconditions. First, given w(y), individuals maximize income net of educationcosts with respect to y. Thus they maximize w(y) c(y,n) by setting

w ( y) = c y( y, n ) .

This holds for all n. The second order condition, w ( y) c yy( y, n )< 0 musthold.

The second condition is that employers experience in the market over

time must be consisten t with th e offers they are m aking, so that for all n ( y) = s( y, n ) .

Ignore the second-order conditions for the moment. Since by assumption

421

16 This section is somewhat technical. It is not difcult mathematics but does require a generalknowledge of differen tial equation s and th e calculus of variations. It is not essent ial for th ose in-terested in learning the basic properties of signaling equilibria and can be skipped by readerswho ar e willing to accept th at the pro per ties cited in th e earlier exam ples sur vive in th e gene ralcase. The second best optimizing calculations are of some interest in interpreting what is goingon in the market.17 Much of the material in this section is taken from parts of the the paper, Competitive andOptimal Responses to Signals: An Analysis of Efciency and Distribution, by Michael Spence,

Journal of Economic Theory, 1974.


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sn > 0, one can in principle solve the equation above for n in terms of w andy, say n=N(w,y). Substituting in the rst order conditions, we have

(1) ( y) = c y( y, N ( , y) ) .

This is a rst-order, ordinary differential equation. It has a one-parameterfamily of solutions that do n ot cross each o ther. In principle each member of

this one-parameter family can be part of a market-signaling equilibrium. 18If c(0,n) = 0 for all n and if c yn < 0, then assuming c y > 0, c(y,n) will be de-

clining in n and that combined with an upward sloping w(y) (without whichno on e would invest in education) , ensures that y is an increasing or at least anon-decreasing fun ction of n. Let the n et income function be

N ( y, n ) = w( y) c( y, n) .

This function has the property that N y is an increasing function of n, whichensures that if N y = 0 for a particular value of n, then if you raise n, N y will be

greater than zero and the maximum will occur at a h ighe r value of y.Differen tiating ( 1) with respect to y we h ave

which m eans that the second order cond ition is in fact satised.Without being too formal about it, education or any poten tial signal tran s-

mits information in equilibrium if its costs absolutely and at the margin de-cline as the unseen valued attribute increases. This is a sufcient condition.

In a later section I will show by examp le that it is not a necessary condition .Since for every level of n , in equ ilibrium w( y) s( n , y), differentiating we

have

because s n > 0 and > 0. Th is implies that in equ ilibrium, the pr ivate re turnto education is higher than its direct con tribution to p roductivity, because of the second term in the equation above. This second term is the signaling ef-

fect. It is the part o f the pr ivate retu rn to the investmen t in education that istied to the unobserved level of n. Since w = c y , the above inequality impliesthat s y c y < 0, so th at in equilibrium for all n, th e investmen t in education ishigher than it would be with pe r fect information. If n were observable, thenindividuals would be paid s(n,y), and they would select education to maxi-mize s c by sett ing s y = c y. Note that if sn 0 for all n, th en the signaling ef-fect goes away. The attribute n is still unobservable to employers but indivi-duals make efcient education investment decisions as they have the infor-

422

18 The solution s cannot cross because if they did, then the der ivative at a given poin t in (w,y)space would have two values, which contradicts the statement that w has a well-dened valueF(w,y) at every point . As a consequ en ce, let u s assume th at th e family of solution s is w(y,K) whereK is a parameter. If w K > 0 anywhere, then it is true everywhere, since to assume the oppositewould m ean that the solution s cross.

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mation they need to make the investment decision op timally, and th ey are th eonly ones who need th at information.

As noted earlier, there is a one-parameter family of equilibrium wageschedules that do n ot cross. Let the paramete r be k and denote the schedu lesby w(y,k). As the solutions do not cross, without loss of generality we can as-sume that wk > 0. Net income to the individual is N = w c. Differentiating

par tially with respect to k, with n held constan t, we h ave

(2) N k = wk + ( w y c y) ( y | k ) = wk > 0.

Thus a shift from one equ ilibrium to an other m akes everyone better or worseoff together. The equilibria are or der able by the Pareto criterion. What h ap-pen s as one moves from one equ ilibrium to another is that th e exten t of over-investmen t in th e signal is increasing or declining for everyone. I have to con-fess that at the time, even after discovering that there might be multipleequilibria, I did not expect that there would be such a simple relationship

among them in terms of market per formance.We can use (2) above and the fact that w s to d erive th e e ffect of a shift in

the equilibrium on the levels of investment in education. N = w c = s c.Differen tiating with respect to k, holding n constant and using ( 2) we h ave

Finally, from th e equilibrium condition w = cy, differen tiating with respect tok with n xed , we h ave

This gives us a pretty complete picture of what happens when the equ ilibriumwage schedule shifts up. It becomes atter, inducing lower levels of invest-ment in education. Average wages decline and average education costs de-cline more, so that net incomes rise and everyone is better off. Figure 6 re-pr esents the family of equ ilibrium wage schedules. As they rise th ey becom eatter and the range of investment in education shifts to the left. Figure 6

shows the range of investmen t in education for each of the schedules.As the schedules rise an d become atter, the r anges of investmen t in edu -cation move to the left, that is they fall.

While it is pe rhaps interesting to note that the signaling effect causes over-investment in the signal by the standard of a world in which there is perfectinformation, th at isnt the world we live in . Thus it may be m ore inter esting toexamine the equilibrium outcomes in compar ison to various second -best out-comes that acknowledge that there is an informational gap and asymmetrythat cannot simp ly be r emoved by a wave of the pen .

I will begin by looking at th e maximization of total net income. Here as weshall see, one can achieve the efcient levels of investment in education forall n. Th e required tax in effect undoes the signaling effect. The reason thatth is is a clean case is because a dollar o f net income is of equal value r egard -

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less of the recipien t, we do n ot get into the business of trading off income di-stribution against the objective of efciency. Following this brief analysis, wewill look at social welfare functions that are not linear in net income. For

these, there will be a tr adeoff between efciency and d istribu tion. I will showthat the market outcome is more akin to the outcome of maximizing a con-vex (that is to say anti-egalitarian) social welfare function. This is not a sur-prising result since the function of signaling is to distinguish low- fromhigher- productivity ind ividuals, which r esults in a m ove away from egalitarianoutcomes. I do also want to emphasize at th e ou tset that the purpose of thisanalysis is not to p ropose or promote taxing or subsidizing signals but rath erto help shed additional light on the properties of the competitive equili-brium.

MAXIMIZING TOTAL NET INCOME

We will assume th at the unobservable attribute n is distributed in th e popu la-tion according to the density function f(n), and the cumulative distributionwill be represen ted by F(n) . The total net income then is

Here N ( y, n ) = w( y) c( y, n ) as before. The constraints are two. Individualschoose rationally so th at w = c y and to tal gross wages have to equal total pro-du ctivity or value gen erated to emp loyers:

424

Figure 6

(3) .


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Our goal here is to select a schedule w(y) that causes education choices y(n)for each n , that maximize total net income, net that is of education costs. Thisis actually pretty straightforward. Using (4), we can replace w(y) in (3) and

choose y(n) to maximize

Th is is the simp lest form of a calculus of variations prob lem. 19 The solution isa schedule y(n) that causes s y = c y , for all levels of n. Th e reader will note th atthe implied levels of investment in education are those which would occur if n were observable, that is in the hypothetical world of perfect information.They are therefore also as a result the efcient levels of investment in educa-

tion for each n : edu cation is invested in u p to th e poin t where at th e marginits direct con tribution to p roductivity ( the hu man capital effect) is equal tothe marginal cost. That is to say, to maximize net income, the wage scheduleis set so as to rem ove th e signaling effect, though education still carries the in-formation an d acts as a signaling. The p rivate par t of the r eturn to the signal(which h as to do with distinguishing one individual from anoth er an d h encefalls into the zero-sum/ red istribution aspect of the market mechanism) issimply removed by the optimal tax or wage schedule. To nd the requiredwage schedu le, you inver t the optimal schedu le y(n) to r( y) , substitute th at in

the optimizing cond ition w ( y) = c y( y, r ( y)) , and integrate to get

The parameter w(0) is then set to cause the breakeven condition (4) to bemet. Individuals are likely being paid their productivity in the market place,s(y,r(y) ) . Therefore one can get to th e desired result by imposing a tax/ sub-sidy on education of t( y) = s(y,r( y) ) - w(y) whe re w(y) is dened by (5) above.If you differentiate t( y) with respect to y you have

t ( y) = s y + snr w = s y c y + snr = snr .

The last term in the equation is the signaling effect at the margin of a changein education. This then says that th e op timal tax at the margin is equal to th esignaling e ffect. It takes that par t of the p rivate r eturn to education away, leav-ing on ly the hu man capital effect or the direct con tribution to p roductivity.

On e can also achieve th e same result with an income tax. Let income be I.Its relation to education is given by I = s(y,r( y) ) . Inver t that function to get

425

(4)

(5)

19

A good tr eatme nt of the calculus of variations can be found in Cou rant . You d isplace y by (n) ,differentiate with respect to and insist that the result be equal to zero for all (n) . This

gives us That in turn implies that s y = c y for all n.

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y=Y( I) and then set th e income tax T( I) equal to I w(Y(I) ) , where againw(y) is dened by (5) . Ind ividuals will select income so as to maximize I T( I) c(Y( I) ,n) = w(Y( I) c(Y( I) ,n) , by setting w = c y which is the desired result.

20

It is interesting for those of us who live in the United States that to a rstcrude app roximation, our education costs are roughly similar in form to theefciency-inducing tax schedule. Lower levels of education are subsidized

and th ese subsidies decline at college an d u niversity levels because of the ex-istence of a large private sector in h igher education .

SECOND-BEST OPTIMA WITH ASYMMETRIC INFORMATION

We turn now to a br ief examin ation of the effects of investmen t in educationand on net incomes of maximizing some social welfare function. As before,N = w(y) c(y,n) is ne t income and n is distributed in th e popu lation accord -ing to f(n) . Let V(N) be th e social value of ne t income of N to any ind ividual.

For the m oment, we will place no restrictions on V(N) o ther th an th at it is up-ward -sloping more net income is better. We will adopt an add itive social wel-fare function

For the moment, we will make no assumptions about the shape of V(N) ex-cept that it is upward-sloping. The objective is to maximize Z subject to twoconstraints. One is that individuals make a rational choice of education

w ( y) = c y( y, n ) .

The second constraint is that gross wages add up to total output or produc-tivity.

Let us suppose that the particular function w(y) that is chosen induces achoice of y = r(n) for each level of the unobserved characteristic. Differen-

tiating N(y,n) totally with respect to n gives

Let N ( n ) = K . Integrating ( 7) with respect to n we h ave

Noting that w = N + c, and substituting in (6), the condition that total wagesequ al total productivity across the whole population , we h ave

426

20 Th is analysis assumes that the re ar e n o adverse incen tive effects of an in come tax in ter ms of the choice between work and leisure . If the re are such e ffects, then taxing income an d taxing thesignal are not equivalent in terms of the outcome.

(6)

(7)

(8)

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The point of all th is is simply to get r id of the fun ction w(y) . Thus we imagineselecting an upward-sloping fun ction y = r( n) to maximize Z. By using (8) and(9) we are assured th at both con straints in th e problem are satised, and the

wage or tax schedule is nowhere in sight. We can calculate it later using theoptimal schedule r( n) by inverting th at schedule to n = h( y) , substituting forn in w = c y and integrating to get the w(y) that induce the signal choices thatsolve the optimizing problem.

The solution to the problem is 21

If V ( N ) is a constant so that V(N) is linear, then the right side of (10) is zero.Th is is the case we h ave just looked at , where total ne t income is maximizedby indu cing the efcient choices of investment in edu cation. It is just a specialcase of the more general problem and in that case, the optimal schedule isdened by s y = c y, for all n .

If V(N) is very concave so that the rst derivative falls rapidly, then at thelimit the on ly thing that matters is the ne t income of the p eople with the low-

est net income. This case is normally called the maximin case. In this case(10) becomes

(11) ( s y c y) f = c yn(1 F ) .

The right side of (11) is positive, which means that investmen t in ed ucation isbelow the efcient and full information level. At the top of the range of n,( 1F) / f appr oaches zero an d the investmen t in education is efcient. 22

The maximin outcome in terms of investment in the signal is also the resultthat would obtain if there were a monopsonist purchaser of labor services.

The reason is that the monopsonist wants to maximize the difference be-tween productivity and gross wages, subject to the constraint that the net in-come of the individuals with the lowest net income not fall below some pre-specied level. Thus the solution to the monopsonists problem is to maxi-mize the n et income of the lowest level and then take it all away by loweringw(y) uniformly across all education levels. Mathematically you are just rever-sing the objective function and one of the constraints.

427

(9)

(10)

21 The der ivation of this is a an oth er application o f the calculus of variations. You displace th efunction r( n) by ( n ), d ifferentiate t he whole thin g with re spect to , and set the r esult equal tozero. After some r eversing of orde rs of integration , you en d up with an equ ation of the form

Since this must hold for all functions ( n ), the optimizing condition is

Q( r( n) ,n) = 0. The app lication of all that t o this case yields cond ition (10) .22 Th is follows from the fact tha t (1-F)/ f is th e recipro cal of the der ivative of -log(1-F).

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The opposite extreme occurs when the welfare function is extremely con-vex so that its slope r ises rapidly. In the limit that would mean valuing on ly thene t incomes of those with th e h ighest net incomes, the maximax case. In thiscase, the op timizing condition ( 10) becomes

(12) ( s y c y) f = c ynF .

In this case, the right side of (12) is negative, implying that like the marketequ ilibrium with imper fect information, the investment in education goes be-yond th e point at which th e direction contribution to p roductivity is reached.At the bottom en d of the range of n, F/ f appr oaches 0 and thus at the lowestlevels the investmen t in edu cation is efcient . 23

For the cases where v(N) is not approaching an extreme (very convex orconcave), one can see from (10) that the right side approaches zero when napproaches its minimum and its maximum values. Provided that f(n) doesno t go to zero at the extremes, this would mean that at th e en d values of n, in-

vestment in education is efcient, that is s y c y = 0. Th is is generally not a fea-ture of the competitive equ ilibrium, and thu s we may conclude th at the com-petitive equ ilibrium is generally not the solution to some op timizing problemwith th is form of social welfare fun ction. It is possible to r ewrite the optimiz-ing condition (10) in th e following form .

where E(*|) is the conditional expected value. For concave functions the

term in the large square brackets on the right is always positive and thus in-vestment in education is below the efcient level for all n except possibly atthe en d points. For convex welfare functions, the d er ivative is rising and th usthe term in square b rackets is negative. Th e levels of investmen t in ed ucationexceed the efcient ones. 24

If you stand back from all this, the general pattern is fairly clear. The mar-ket equilibrium produces overinvestment in the signal because of the signal-ing effect, which is a private benet to th e investor, but yields no social ben e-t as its function is purely redistributive. The redistribution occurs in the

428

23 Th is follows from the fact th at F/ f is the r eciprocal of the der ivative of log(F).24 Th ose familiar with op timal tax pr oblems will recogn ize th e form of this optimizing con dition.In th e incom e tax prob lem, individual welfare measured in dollars is u ( w,l) = wl t ( wl) h( l- l) ,where w is income per hour worked, l is working hours and t(wl) is a tax on income, and l- is aconstant- one can thin k of it as the total time available. Individuals maximize u with r espect to l

by setting w t w h = 0. The govern men t has a reven ue constraint so that If v(u )

is the welfare function the n we want to maximize He re you chan ge the optim izing

sched ule to y(w), get rid of tax function t( y) by no ting that impo se the reven ue con-

straint to determine u (0), and the displace the function y(w) to n d the optimum. The optimum

is achieved whe n Note that if v(u ) is linear, th en

the right side of this equat ion is zero. Work leisure choices are efcien t and the govern men t rev-en ue is raised by a lump sum tax, which doe s not d istort the work-leisure choice.


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direction of increasing the gross and net incomes of those with higher levelsof education and productivity and hence income , the market equilibria tendto look more like the solutions to a second-best optimizing problem with aconvex social welfare fun ction. These, no t surp risingly, are the welfare fun c-tions that weight higher ne t incomes more highly than lower n et incomes.

TH E CASE OF EDUCATIO N COSTS RISING WITH TH E UNSEENATTRIBUTE: SIGNALING COSTS VARYTHE WRONG WAYWITHRESPECT TO PRODUCTIVITY

The stand ard case of signaling in which th e signal has the capacity to sur viveand retain its informational con ten t occurs when there is an un observable at-tribute that is valuable to buyers (in the examples we are looking at, em-ployers) , and the costs of undertaking some activity that is observable are ne-gatively correlated with the valued attribute. The labor market examples,

however, are slightly more complicated in that in that the unobserved at-tribute contributes to the individuals productivity and so does the signal.Thus, whatever the unobservable attr ibute is, it has two sources of value. Oneis the direct effect on productivity and the other is that lowering the costs of acquiring h uman capital also has value. Up to th is poin t we have assumed thatall of this works in the same direction. But it is possible that attributes thatlower the cost of acquiring education might not be those that enhance pro-ductivity or might even be attributes that have a negative effect on producti-vity. We know that if S n = 0, then you can get a signaling equilibrium, though

in that case the signal isnt needed: it simply identies ex post those withlower education costs and hence higher levels of education .

The question I p ose n ow is, can you get a signaling equ ilibrium when c yn > 0,assuming that sn > 0? From the an alysis of equ ilibria, we kn ow that th e secondorder condition for the individuals choice of education is satised only if

Th is mean s that if c yn > 0 one can have an equilibrium

with signaling on ly if So the question is, can that happen? The on ly

way it could happen is if the human capital effect is large in the sense that it

overrides the n egative signaling effect. Th us if s y 0, it certainly cannot h ap-pen. The signal makes no direct contribution to productivity and sends thewrong message about the unobserved attribute. Everyone will set y = 0. Butthe answer to the question posed above is yes, provided th at the h uman capi-tal effect is large enough to override the negative signaling effect. There canbe signaling equilibria in which signaling costs rise with the level of the un-observed a ttribu te th at con tribu tes to productivity. I will demonstrate th is byexample in a moment. Intuitively this should make sense if the human cap italeffect is big enough, because then the attribute th at is of real value to th e in-dividual and to th e employer is the one that dr ives education costs down. Thekey is that the wage schedu le has to be upward-sloping in the signal, and th iscan happen if s y is big enough even if sn > 0. In this context, the correct state-ment about the condition that allows for a signaling equilibrium is that the

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net benet of acquiring the signal has to be positively correlated with thegross effect on produ ctivity. This could h appen in segments of the populationif very talented people face high opportunity costs associated with spendingtime on education.

This case may or may not be interesting from an empirical poin t of view,but it does illustrate that th e mor e general formulation of the conditions for

signaling are in terms of gross and net benets and not just signaling costs. Idid n ot realize th is when I rst worked on signaling equilibria. I though t thenthat the absence of the intuitively plausible negative cost correlation condi-tion would destroy a signaling equilibrium. It is also of potential interest be-cause th e signaling effect is reversed . If c yn > 0, then in equ ilibrium if there issignaling, dn/ dy < 0, and thu s

Therefore the negative signaling e ffect causes the private return to ed ucationto fall short of the social return and hence causes under-investment in edu-cation. 25 It rem ains to show by a constructive example that this can happen atleast in th e th eory, and perh aps also in the world.

We will show that th is can hap pen with an example. Let s( n, y) = ny , and letc( n, y) = n y , where < 1. Following the standard equilibrium analysis, wehave

(13) w = c y = n y 1 .

In addition w = ny , or n = wy

. Substituting in (13), we have the differential

equation in w(y) that den es the equ ilibrium wage schedules

w w = y( 1 ).

On e of the solutions to th is differential equation is 26

where K is a constant. If you th en proceed to dete rmine equ ilibrium choices

of education, they are given by

where T is ano ther constant. The second order cond ition is satised an d th isall works if dy/ dn < 0 and th is will in fact be th e case provided that < 1, and < . That is to say, signaling occurs and there is a separating equ ilibrium, if the e lasticity of productivity with r espect to education is larger than the e lasti-city of education costs with respect to education. Or in simple ordinary lan-

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25 I thou ght Professor Gary Becker who se pioneer ing work in hu man capital, ho usehold p rod uc-tion functions and much more, m ight ap preciate this result.26 As this is simply an illustrative exam ple, th ere is not m uch point in studying all th e eq uilibria inthe example.

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guage, education is productive enough to justify its costs and override thenegative signaling effect.

Another way to think about these relationships and this case is to changethe un observed variable from n to t = n n , where n is the h ighest level of n .In effect this denes the unobserved characteristic to be that which lowerseducation costs. Now investmen t in education will rise with t , provided there

is a separating equilibrium, because c yt < 0. But the effect of t on productivityis now negative, and if that effect is strong enough relative to the human ca-pital effect, then that will prevent a separating version of the signaling equi-librium. The reason is that unless the human capital effect is large enough,the signaling effect will cause the wage function to be downward-sloping in y.There will be a pooling equilibrium at y = 0 and that of course entails in thehypoth etical example, reasonably dramatic under-investmen t in the poten tialsignal even though it is productive human capital.

TH E INFORMATIO N CO NTAINED IN TH E SIGNAL CAN IMPROVEPRODUCTIVITY

It should be noted that the information carried by the signal can be produc-tive itself. This will occur if there is a decision that is made better or withgreater efciency, with bette r information. In the job market con text, onecould build th is in as follows. Let productivity be v(n,y,d) where d is a decisionthat the employer makes. It could be a decision about what type of job to as-sign to someone or a decision about required training. One can interpret th e

preceding analysis then, at least for equilibria in which the signal carries in-formation in equilibrium, as s(n,y) being th e m aximum of v(n,y,d) with r e-spect to d for each level of n and y. Note th at th is comp onen t of value is dif-ferent from the human capital effect. Unlike the human capital effect, thiselement of value only exists if the signal is carrying information in equili-br ium. You would lose it in a par tial or comp lete pooling equilibrium even if that equilibrium included investmen t in education .

In situations like the Lazear model that we examined earlier, in whichthere is both p ooling and separ ating componen ts of the equilibrium, the ad -

dition of a decision that is made better with information will cause the valueof the signal to rise, or more accurately the net benets to rise. Those netbenets are passed on to individuals by virtue of competition on the employ-er side. Thu s the relative sizes of the separating and p ooling compon ents of the equilibrium will shift in favor of the separating par t. More peop le will optto go to the rms that incur the expense of mon itoring and learning produ c-tivity over time.

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TIME AND THE ALLOCATIO N O F TIME AS A SIGNAL AND ASCREENING DEVICE 27

The pr inciples that govern the survival of signals in markets can be app lied toother contexts. The kind of mixed or imp er fectly aligned incentive structurethat char acterizes markets is common in many situations. In particular, situa-tions in which ind ividuals have an un known or imper fectly perceived level of interest in someone or something are ubiquitous. In these situations, onefrequen tly observes that time spent by the individual is taken as a signal of in-terest. Literature and everyday experience suggest that the allocation of timeis a ub iquitous and persisten t signal sometimes used deliber ately as a screen-ing device.

The persistence of spen ding time as a signal of interest in someth ing is a re-ection of the fact that time is in shor t supply and everyone knows it. There isa non- zero shadow pr ice on time, and hence th e use of it in pu rsuit of someperson, goal or interest must mean that some implicit net benet test hasbeen passed for those who spen t the time and failed for others who did not.

The use and inte rpretation of time as a signal is complicated an d en richedby the accurate observation that there are both perceived and actual differ-ences across people in the shadow pr ice on time. These d ifferences are u sedto inter pre t the signal. The allocation of a small amoun t of time by an indi-vidual with a high perceived shadow price on time has the same weight as alarger allocation of time by an ind ividual with a lower p erceived shadow pr ice.As an example, those who have had visible leadership positions in virtuallyany organization know that the events they attend are taken as signals of in-terest and support, an d conversely, when they do n ot show up , they delibe-rately and sometimes inadvertently signal a lack of interest, support or en-thusiasm. It is noteworthy that the informational content of these commonsignals is not necessarily related to wheth er the ind ividual ( the signaler ) is ac-tually needed or has a role at th e even t. In fact, having a role at th e even t candilute the signal because it complicates the interpretation of the reason forthe leaders presence.

Time is also routinely used as a screening device or as part of the price of adm ission to even ts at which the number of places is in shor t supply. For en -tertainmen t and sporting even ts it is common to nd th at a mixed system thatinvolves both pr ice an d time is used to allocate the limited number of placesat the event. The question that I address briey in this section is why themixed system would be u sed in p referen ce to th e pure p rice system. Th e in-tuitive answer is that willingness or ability to pay may be a poor signal of realinter est in the even t, not withstand ing the fact that failure to u se the price sys-tem is inefcient for two reasons. 28 The mixed system allocates places or ac-

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27 Part of this section is based on a nearly unknown (probably for good reason) paper by theauthor, Time and Communication in Economic and Social Interaction, Quarterly Journal of

Economics.28 There are pe rformers who want p eople with a gen uine inter est, and not just tho se with an in-tere st and h igh income s at the events, and who feel that th e qu ality of the e ven t is affected b y in-teraction between th e per former and amon g members of the audien ce.


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cess to th ose who would (and often d o if there is a second ary market) sell theplace to someone with a higher willingness to pay when that happens bothparties are better off. Second , the time that is used to acqu ire the p laces is adead-weight loss under most circumstances. Notwithstanding the clear prob-lems with the failure to use the price system, the use of the mixed system isvery common, and must mean that some objective is being pursued that is

no t respectful of the cur ren t distribution of income. For if the d istribution of income is viewed as optimal, it is hard to con jure up a rationale for incurr ingthe costs of the mixed system mentioned above.

I am going to use a simple example to look at the effects of using a mixedsystem for this class of resource allocation problems. It is not meant to be ageneral model, but rather to suggest what parameters are important in de-termining the per formance characteristics of the ou tcome.

Let the value attached to an event by an individual be . One can think of this as a dollar valuation based on a roughly equal distribution of income, or

you can think of it as a measure of value in a pu re time system. What it is notis a current willingness to pay for access to the event. It is uniformly distri-buted in the population, on the interval [0,1]. The value of money is . Italso is distributed un iformly in th e popu lation on the interval [0,1]. The as-sumption about the value of time at the margin is that it is equal to G . If the parameter a is zero, th en it is a constant, if a > 0, then the marginal valueof time is negatively correlated with the marginal value of income, and theconverse is also true. The fraction of the population that can have access tothe even is N, and we will assume th at N < 0.5. Finally let p be the monetar y

pr ice for access to the even t and let T be th e time price.The set of people who attend the event is the people with combinations

of( , ) that satisfy the inequality

p (G )T = ( p T ) GT 0.

Let S = GT an d R = p aT . Then the inequality above becomes

R S 0.

Let A = {( , ) ) : R S 0}. Then because of the xed supply, the area of

the region associated with A has to be N.Figure 7 is useful in analyzing the cho ices. Each of the lines inside the unitsquare represen t combinations of R and S, so that the area above the line isN. The line th rough th e origin rep resents the pure pr ice system. As the timeprice T is raised, S rises and R falls. It is easy to see without a complicatedproof that as T rises, the average value of is rising while the total numbersstay the same, so that the total gross value to consumers is rising. This conti-nues until the line goes at, at which point R = 0. It is also clear that the in-creases are largest for low levels of T or S, because initially one is trading lowlevels of for much higher levels.

It is possible that the mixed time and pr ice system is commonly used simp lyto achieve this result, namely allocating the spots to those who attach thehighest value to them, and that whoever makes these decisions doesnt think

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about or care about the dead weight cost of the time component of the p rice.

But p erhaps a more inter esting question is whe ther th e u se of the mixed sys-tem ever increases the net benets. We will assume that the dollar revenue isdistributed in a n eutral way so th at pr ice comp onent of the access charge ge-nerates neither net benets or costs. The time component of the accesscharge however generates a dead weight cost.

The n et benets are

The second two term s are th e dead-weight cost of the time comp onent of theaccess charge. We h ave alread y observed that th e rst term, th e gross benets,are a rising function of S = GT, reecting the use of time as part of the allo-cation mechanism. This is true independent of the sign or size of the para-meter a. Moreover, one can see from Figure 7 that starting at S = 0, thegains from increasing S, or T, are largest at the start (that is near S = 0) be-cause as the lines rise an d r otate right, low average levels of the valuation pa-rameter are being knocked out and high ones are being brought in. Thedifference between the exits and the entries steadily declines as S increases,and the lines become atter.

This brings us to the question of whe ther th e n et ben ets also increase atleast over some range as s increases. It is somewhat ted ious but not d ifcult toshow that in th is examp le that the slope of gross ben ets at S = 0 is

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Figur 7


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The der ivative of net ben ets at S = 0 is 29

If the parameter a is zero, then one can see that in this example, the dead-weight costs outweigh th e gain in gross benets. However if a is positive an dlarge enough, th en the ne t ben ets rise at least over some r ange. Why shouldthis occur intuitively? Here price is a badly awed signal of value. Time, if itsmarginal cost is uncorrelated with price, is a better signal of value, but notgood enough to overcome the dead-weight cost. But if the marginal cost of time is negatively related to th e marginal value of income, th en the combina-tion of price and time is a much better signal of value. And if marginal timecosts fall quite rapidly as the marginal value of income increases, then thecost of the time that is need to screen out the h igh income-low value purchasersis relatively low.

To illustrate, Figure 8 shows the value o f net benets as a fun ction of S, fora case in which G = 2, N = .25 and a = 14. Here th e r eader should ignore th ehigher levels of S, as they are associated with negative prices. Above a certainlevel of S, or time cost, the model starts redistributing income, which iscoun ter to the spirit of the analysis. 30

To summarize, while this simple example hardly disposes of the issue of how markets go about t rying to solve complex allocation prob lems in a worldof imper fect information , it does, I hope, suggest that th e kinds of mixed sys-tems that we observe in reality may be quite innovative solutions to theseproblems and p robably should not be dismissed in all cases as mistakes mad eby people who dont understand the virtues of the price system. There mayeven be more interesting and less wasteful alternative currencies, such ashour s spent in public service work, that could be used in con jun ction with thepr ice system to allocate h ighly valued access to imp ortant even ts. While suchalternatives might not be as effective as time in und oing the effect of the pricesystem, they may also h ave considerably less dead weight cost associated withthem.

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29 Th ese results follow from th e following facts. For R + S > 1, gross ben ets are

and th e dead weigh t time cost is Similar ly,

whe n R + S < 1, which occurs as S beco me s larger, R = 2(1 S N ), gross benets are (1/ 2)[( 1

S2) RS ( R 2 / 3], and the dead weight cost of the use of time is Whe n R + S = 1, the

line that separates those who are adm itted from those who are n ot goes through th e point (1,1)which is why the form ulae shift at th at po int.30 Th e existen ce of secon dar y mar ket will to some exten t red uce th e ben et of the mixed systembecause it will draw into to primary allocation system entrepreneurs who are there to use theirrelatively low valuation o n tim e to m ake an ar bitrage pr ot. Th ey will bid up the price re quiredto clear the primary market and hence drive some of those with high gross valuations of theevent from that m arket.

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TH E INTERNET AND TH E CHANGING INFORMATIO NAL STRUCTUREOF MARKETS

The proposition that th e inter ne t, or more accurate ly its relatively recen t ac-cessibility to a broad spectrum of users, has changed the informational struc-

ture of many markets, industries and economies, is probably no t con trover-sial. One could argue about how rapid the change has been from quitesudden on the on e hand to a gradual evolution from the telegraph ( whichwas the rst time that communication over d istance d id no t require th e ph y-sical movement of something an d h ence became in a certain sense n early in-stantaneous) th rough rad io, the telephone, television, fax etc. on the other. Ithas also been easy for some to d ismiss all of this as a fad, re lying on the boom-and-bust cycle that we h ave just observed. But in my judgmen t th at would bea mistake. We know from the economics of information that there are periods

in which en ough has changed so that we operate in a data-free environm ent,that is, one in which there is temporarily little or n o re levant data to constrainexpectation. The data trickle in and th e be liefs and expectations start to lineup with reality again.

The more fundamen tal point is that investors and other s were probably notwrong about the ultimate effect of this technology on markets and the eco-nomy, but almost everyone overestimated the speed with which individualsand organizations change their behavior, and we also underestimated theamount of difcult technical infrastructure that needed to be built in orderto h ave the foreseen ou tcomes becom e a reality. We would un doubted ly havebeen better off if we had reminded ourselves of the important work of thelate Zvi Griliches on the diffusion of innovation, as that would have causedsome questioning of the assumption that what very intelligent people can

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Figure 8


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foresee will come about quickly. I wanted also to recognize Zvis work as anearly and very important example of behavioral economics.

Neverth eless there are power ful forces driving the ou tcomes and changingthe informational structure of markets. Three of the most important areMoores law, Metcalfes law and the d ramatic reduction of the n oise to signalration in ber optic cable resulting in very large expan sions in the thr ough-

pu t capacity of these p ipes measured in signal p rocessing terms. Moores lawis well-known. It is an emp irical observation that th e n umber of tran sistors ona chip doubles every eighteen months to twenty-four months. To a rst ap-proximation this has produced roughly a 10-billion-times cost reduction inthe rst fty years of the compu ter age, dated from abou t 1950. Or to pu t itanother way, th ings that were imaginable but un imaginably expensive in 1950are essen tially costless today. Economic h istor ians will be better able th an I tosay whether th ere were comparable per iods of change in th e past with respectto costs of doing something importan t.

Metcalfes law states that the value o f a ne twork to th e ent ities attached to itis proportional to the square of the number of connected entities. 31 In eco-nomic terms this probably mean s that th e value and hence th e speed of con-necting accelerates as the nu mbers increase. Th is is sometimes referr ed to asthe network effect.

From signal processing theory we know that the capacity of a channel isproportional to the log of one plus the ratio of signal to noise. Throughscientic and technical advances (in addition to being able to use lasers withmultiple wavelengths), the noise generated when th e light reects off the side

of the ber optic cable is being reduced dr amatically, causing very large in-creases in data rates on existing ber. In economic terms this translates intolarge cost r eductions in providing bandwidth.

These three effects interact with each other over time to produce accele-rating economic effects. Much of the recent econ omic impact of the intern etis associated with reducing transaction costs of a variety of kinds. I will returnto th is subject in a moment. For m ost of these effects, one needs no t just com-pu ting power, but also a reasonably ubiquitous re liable network that h as stan-dardized protocols, is more or less always on. After all, transactions involve,

almost by den ition , more than one par ty, though increasingly one or moreof the parties are machines. It is interesting and n ot surpr ising ther efore, thatfor the rst forty years of the p roliferation of compute rs, no t mu ch measur-able productivity increase was detectible in macroeconomic data. In the pastten years, the per iod in which th e expansion of the n etwork occurred, the reare noticeable measurable productivity gains. It seems very likely that th e p ro-

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31 This should be thought of as an empirical regularity as well. It can be derived as follows.Supp ose th at the value a person to each oth er pe rson on a network is x. If there are n people onthe n etwork and we add an other p erson, the n p eople experience added value of nx and the n ewper son exper iences value of n x. If V(n ) is th e value of a n etwork, bein g the sum of its value to allconnected people, then V(n + 1) = V(n) + 2nx. Therefore V(n) = 2x[1 + 2 + ... + (n-1)]. UsingGausss form ula for th e sum of the integers from one to n , we h ave V(n ) = 2xn( n-1), which is thequadratic relationship that Metcalfes law asserts.


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ductivity gains (the ones we can detect) are associated with reduced transac-tion costs, improved per formance of markets, the ability to create new mar-kets that were too expensive to create without the techn ology, and the im-portant ability to take time and cost out of the coordination of economicactivity, inside the rm and in the supply or value-added chain. That is whatthe network, with h igh and reliable capacity, with growing number s connec-

ted to it and with enough computing power to run the endpoints and thenodes has permitted. It is a cumulative effect that we are just beginning tosee.

Building mod els in economics is an ar t and a science. The science part con -sists of determining analytically the consequences of the assumption that cre-ate the structure o f the model. The art consists in deciding what to pu t intothe structure and what to leave out. Putting in too m uch m akes the models in-tractable and o f little use in illuminating the de terminants of market per for-man ce. Putting in either too little or th e wron g structural features makes the

results, though tractable, uninte resting. A natu ral consequen ce of this is thatparameters that dont change much from market to market or over time tendto be supp ressed, as a matter of good practice in app lied m icroecon omic the-ory. I mention all of this because it is possible and even likely that some para-mete rs related to search costs, tran saction costs, acquiring information, andgeograp hy have shifted reasonably rap idly in th e past few years ( or ar e in th eprocess of shifting), as a result of the increasing speed, ubiquity and connec-tivity of the intern et. Th is potential shift in parameters creates the opportu -nity to revisit aspects of the informational structure of markets and organiza-

tions, and in a sense, reintroduce the suppressed parameters as they havebecom e in teresting again. I would like to con clude th is essay by suggesting afew areas in which such an inqu iry might yield some in teresting r esults.

On e might h ave the impression from a rst course in economic theory thatmarkets combine supply and demand curves to produce prices, quantitiesand to determine who buys the good. There is nothing wrong with that view.But in add ition , a market economy per forms lots of other fun ctions. Poten tialbuyers and sellers need to nd each other. Often buyers and sellers need toacquire information ab

a. michael spence - nobel lecture

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