SI G NALI N G I N RETR OSPECT A N D T HE I NF OR- MATI ONAL STR UCT URE OF MARKETS Prize Lecture, Dece mber 8, 2001 by

A. M IC HAEL S PE NCE 1 Sta nford Busi ness Sc hool, Sta nford U niversity, 518 Me morial Way, Sta nford, C A 94305-5015, US A.

INTRODUCTION W he n I was a graduate stude nt i n eco no mics at Harvard, I had t he privilege of serving as rapporteur for a faculty se minar in the then-ne w Kennedy Sc hool of Gover n me nt. A mo ng ot her disti nguis hed sc holars, it i ncluded all of my thesis advisers, Kenneth Arro w, Tho mas Schelling, and Richard Zeck- ha user. I n t he c o urse of t hat se mi nar t here were disc ussi o ns of statistical dis- cri mi natio n a nd ma ny ot her subjects t hat relate to t he i nco mplete ness of i n- for mation in markets. One of my advisers ca me in one day with the strong suggestion that I read a paper he had just read called “ The Market for Le mons” by George Akerlof. 2 I al ways di d w hat my a dvisers tol d me to do a n d he nce follo we d u p i m me diately. It was q uite electrifyi ng. T here we all fo u n d a wo n derf ully clear a n d pla usible a nalysis of t he perfor ma nce c haracteristics of a market with inco mplete and asy m metrically located infor mation. That, co mbi ned wit h my puzzle me nt about several aspects of t he discussio n of t he consequences of inco mplete infor mation in job markets, pretty much lau nc hed me o n a searc h for t hi ngs t hat I ca me to call sig nals, t hat would carry i nfor matio n persiste ntly i n eq uilibri u m fro m sellers to b uyers, or more ge nerally fro m t hose wit h more to t hose wit h less i nfor matio n. 3 T h e iss u e, of course, was t hat sig nals are not terribly co mplicated t hi ngs i n ga mes w here t he parties have t he sa me i nce ntives, i.e. w here t here is a co m mo nly u nder-

1 T his paper is based o n a lecture delivered by t he aut hor i n Stock hol m o n t he occasio n of t he 1 0 0 t h a n niversary of t he fo u n di ng of t he Nobel Prize. I wo ul d like to t ha nk my fello w reci pie nts of t he Nobel Prize i n Eco no mics t his year, Professor George Akerlof a n d Professor Jose p h Stiglitz, for t heir work a nd t heir i nspiratio n, a nd my t hesis advisers, Professors Ke n net h Arro w, T ho mas Schelling and Richard Zeckhauser whose ideas and guidance got me launched on the study of market structure (particularly i nfor matio nal structure) a nd perfor ma nce. My colleagues Profes- sors Ed ward Lazear a nd Mark Wolfso n gave me a great deal of co nstructive i nput. I also o we a great debt to Professors Ja mes Rosse a nd Bruce O we n, wit h w ho m I lear ned a nd taug ht i ndustrial organization and applied microecono mic theory at Stanford. It was a great group of young p e o pl e a n d a w o n d erf ul ti m e t o b e i n t h at p art of t h e fi el d. 2 George Akerlof, “ The Market for Le mons: Qualitative Uncertainty and the Market Mechanis m, Q uarterly Jo ur nal of Eco no mics . 3 I believe is was R o bert Jervis w h o i ntr o d uce d t he ter ms “i n dices” a n d “sig nals.” I n dices are attri- b utes over w hic h o ne has no co ntrol, like ge n der, race, etc. T hi nk of t he m as u nalterable attri- b utes of so met hi ng, not necessarily a perso n. Sig nals are t hi ngs o ne does t hat are visible a n d t hat are i n part desig ne d to co m m u nicate. I n a se nse t hey are alterable attrib utes. I t ho ug ht it was a us ef ul s et of disti n cti o ns a n d t er mi n ol o gy a n d I still d o.

4 0 7 stood desire to co m municate accurate infor mation to each other. Even in t hat case (so meti mes called t he pure coordi natio n case) ho wever, t here are pote ntial proble ms of c hoosi ng a mo ng eq uilibria as ill ustrate d i n Sc helli ng’s brillia nt a nalysis of t he use of f ocal p oi nts a n d c o ntext ual i nf or mati o n t o s olve co m munication/coordination proble ms when the parties have been de- prive d of t he a bility t o c o m m u nicate directly. 4 I n markets w here t he iss ue is of- te n u ndetectable or i mperfectly detectable quality differe ntials, t he alig n- me nt of i nce ntives is ty pically i m perfect a n d t he i nce ntive of t he hig h- q uality pro d uct o w ners to disti ng uis h t he mselves a n d t he i nce ntive of t he lo w-q uality o w n ers t o i mit at e t h e si g n al s o as t o o bs c ur e t h e disti n cti o n is f airly cl e ar. Of co urse t here is more to it, as o ne nee ds to k no w s uc h t hi ngs as w ho is i n t he market persiste ntly a n d he nce w ho has a n i nce ntive to establis h a re p utatio n t hroug h repeated plays of t he ga me. I a m goi ng to devote a good portio n of t his lect ure to t hese iss ues, i n a se nse to revisit sig nali ng, a n d t he n t ur n to so me ot her aspects of t he i nfor matio nal structure of markets t hat are raised by t h e p ar a m et er s hifts c a us e d by t h e pr olif er ati o n of t h e i nt er n et as a c o m- mu nicatio n mediu m i n t he past fe w years. 5 I was aske d rece ntly by a so me w hat i ncre d ulo us q uestio ner (act ually a jo ur- nalist) whether it was true that you could be a warded the Nobel Prize in Eco no mics for si m ply notici ng t hat t here are markets i n w hic h certai n parti- cipants don’t kno w certain things that others in the market do kno w. I t ho ug ht it was pretty f u n ny. It was as if t his ha d so me ho w bee n a closely g uar d- e d secret u p u ntil a b o ut 1970, at least i n ec o n o mics. I clearly ca n n ot s peak f or t hose w ho make t he decisio ns about t he Nobel Prize, but I suspect t hat t he correct a ns wer to t hat q uestio n is no. W hat di d blosso m at t hat ti me was a se- rious atte mpt by many talented econo mists to capture in applied microeco- no mic t heory a w hole variety of aspects of market structure a nd perfor ma nce. T hat work pro d uce d a partial mel di ng of t heory, i n d ustrial orga nizatio n, la- bor eco no mics, fi na nce a nd ot her fields. A n i mporta nt early part of t hat ef- fort was t he atte mpt to capture i nfor m atio n al as pects of market str uct ure to study the ways in which markets adapt, and the consequences of infor ma- tio nal gaps for market perfor ma nce. T herefore, i n a ns wer to t he questio n, we noticed t hat t here are ma ny mar- kets wit h i nfor matio nal gaps. T hese i nclude most co nsu mer durables, virtual- ly all job markets, ma ny fi na ncial markets, markets for various types of food and phar maceuticals, and many more. These infor mational gaps were widely ackno wledged and those of us who taught applied microecono mic theory, freely ad mitted that these gaps might change so me of the perfor mance cha- racteristics, not to me ntio n t he i nstitutio nal structure, of markets i n w hic h

4 T h o m as S c h elli n g, T he Str ate gy of Co n flict . 5 I n eco no mics a n d ot her social scie nces, mo dels abstract fro m reality a n d foc us o n str uct ural fea- tures of orga nizatio ns or markets t hat drive outco mes. T here are e mbedded a nd usually u nstated para meters t hat are not t he foc us of atte ntio n, as t hey do not nor mally c ha nge. W he n I refer to pa- ra meter s hifts ca use d by t he i nter net, I mea n c ha nges i n str uct ural ele me nts t hat for merly di d not change. For exa mple, most markets have geographic di mensions and boundaries that are i mpli- citly u nderstood a nd are not t he focus of atte ntio n. T he i nter net ho wever, has moved t hose bou nd- aries by collapsi ng ti me a nd dista nce i n t he i nfor matio n/co m mu nicatio n di me nsio ns of markets.

4 0 8 t h ey a p p e ar. B ut I t hi n k it is f air t o s ay t h at w e di d n ot h av e m u c h syst e m ati c knowledge based on theory of what those changes might be. And so we thought that applied microecono mic theory deserved an atte mpt to build t hese i nfor matio nal c haracteristics i nto mo dels t hat ca pt ure t he str uct ure a n d perfor ma nce of t hese markets wit h reaso nably accurate assu mptio ns about t he ex a nte i nfor matio nal co n ditio ns. It was a very exciti ng ti me for all of us w ho w er e i nv olv e d. O n e of t h e w o n d erf ul as p e cts of r e c eivi n g t his a w ar d is t h at it has triggere d i n me, a n d I t hi nk ot hers, fo n d me mories of t he se nse of disco- very a n d excite me nt. A n d i n t hat co ntext, I wo ul d like to take t his o p port u ni- ty to express my ad miratio n a nd deep gratitude to t he ma ny exta nt you ng col- leagues with who m I worked and shared these ideas. They should, and certai nly i n my mi n d, do s hare i n t he recog nitio n triggere d by a war di ng of t he disti nguis hed Nobel Prize for w hat was acco mplis hed duri ng t hose years. T h e pl a n f or t his l e ct ur e is as f oll o ws. T h e ov erri di n g g o al is n ot t o l o o k t o o stupid to t he next ge neratio n of stude nts a nd sc holars. More seriously, fol- lo wing the advice that my advisers once gave me, I will first discuss the si mplest model t hat I ca n devise t hat illustrates i n reaso nably ge neral for m t h e d e fi niti o ns a n d pr o p erti es of si g n ali n g e q uili bri a. 6 N e xt, w e will all o w t h e signal (in this case education in the job market context), to contribute dir e ctly t o t h e pr o d u ctivity of t h e i n divi d u al as w ell as f u n cti o ni n g as a si g n al. Follo wi ng t hat, t he paper exa mi nes a market i n w hic h t here is sig nali ng a nd b ot h s e p ar ati n g a n d p o oli n g i n t h e e q uili bri u m. T he sectio n after t hat exa mi nes a fairly ge neral partial eq uilibri u m mo del of sig nali ng a n d disc usses co m petitive eq uilibria a n d certai n ki n ds of “o pti mal” re- s p o nses t o sig nals. 7 T hose w ho are more i nterested i n t he ge neral idea a nd less i n t he ge neral case ca n ski p over t his sectio n wit ho ut risk of missi ng a ny i m por-

6 Jo h n vo n Neu ma n n, w ho deserves to be o n a s hort list of t he greatest mi nds of t he 20 t h ce nt ury, is re porte d to have sai d t hat yo u do n’t u n dersta n d a t heory or a n abstract str uct ure u ntil yo u have see n a nd worked t hroug h hu ndreds of exa mples of it. Eve n if he did n’t say t hat (a nd I t hi nk he did), I agree with it. Fe w people can say that they had the idea that calculating machines did n’t have to be hard- wired, t hat you could store i nstructio ns i n me mory a nd t he n execute t he m i n order: t hat is, you could build a progra m mable co mputer. 7 These opti mal responses proble ms are selection proble ms. In insurance markets, they can be moral hazard or adverse selectio n proble ms, a nd i n ot her co ntexts t hey are age ncy proble ms or opti mal taxatio n proble ms. You wa nt to co nfro nt people wit h sc hedules t hat cause t he m to make appropriate c hoices a nd i n so doi ng to reveal t he mselves a nd t he i nfor matio n t hat t hey privately hol d ex a nte. Ge nerally t he res ults are belo w w hat is ac hievable wit h perfect i nfor matio n, beca use wit h perfect i nfor matio n, you do n’t have to worry about t he revelatio n proble m. T hese proble ms have t he eco no mic a nd mat he matical properties of seco nd-best taxatio n proble ms, of w hic h per- haps the first to be analyzed was the “opti mal” inco me tax proble m, by the Reverend Ra msey, t houg h t he a nalysis see med to have largely escaped notice u ntil Ji m Mirrlees, To ny Atki nso n, a nd t he n ot hers re discovere d it. T he i dea is t hat t he taxi ng a ut hority ca n not directly observe a n i n di- vidual’s ear ni ng po wer (t hi nk of it as maxi mu m pote ntial hourly wage) so t hey tax w hat t hey ca n see a nd mo nitor, w hic h is i nco me. T his produces a disi nce ntive wit h respect to work t hat could have bee n avoi de d if t he tax were base d o n ear ni ng pote ntial rat her t ha n act ual ear ni ngs. T he se- co nd best opti mizi ng calculatio n t he n trades off i n t he most ef ficie nt way disi nce ntives for work a nd t he need to meet a reve nue goal for t he taxi ng aut hority. T he mat he matics of t hese proble ms are so meti mes referred to as opti mal co ntrol proble ms a nd so meti mes t he calculus of variatio ns. I n later work, Hol mströ m applies t hese models to a labor setti ng, i ncorporati ng Wilso n’s work o n sy ndicates (opti mal risk-s hari ng). Here t he seco nd-best opti mizi ng calculatio n sacri fices opti mal risk- s hari ng e m ployees a n d o w ners to provi de i nce ntives for e m ployees to work dilige ntly.

4 0 9 ta nt pro perties. Ho wever, t his is t he sectio n i n w hic h t he for mal relatio ns hi p be- t wee n sig nali ng models a nd models of self-selectio n a nd opti mal taxatio n wit h i mperfect i nfor matio n is exa mi ned, a nd t hat may be of i nterest to so me readers. I als o w a nt t o i ntr o d u c e i n t h at c o nt e xt a n e w p ossi bility t h at I dis c ov er e d o nly r e- ce ntly a n d t hat is t hat o ne ca n have a sig nali ng eq uilibri u m i n w hic h t he costs of t he sig nal a p pear t o vary wit h t he u nsee n a bility c haracteristics i n t he wr o ng way. T hat is t o say t he c osts of e d ucati o n (a bs ol utely a n d at t he margi n) rise wit h a bi- lity, or more ge nerally, wit h t he u nobserve d attrib ute t hat co ntrib utes positively t o pr o d uctivity. F or t he m ost part, I will use t he j o b market case f or ex p ositi o nal purposes a nd co m me nt o n so me ot her sig nali ng situatio ns more brie fly later i n t h e ess ay. T h er e is a ris k i n usi n g j o b m ar k ets t o ill ustr at e si g n ali n g. I n or d er t o illustrate t he fu nda me ntal properties of sig nali ng models, I have stripped a way ot her feat ures of t he market, partic ularly i n t he si m pler mo dels. B ut I have no- tice d i n t he past t hat t here is a te n de ncy for t he si m pler mo dels set i n t he job market co ntext, to co nvey u ni nte nded messages suc h as (1) educatio n does not co ntrib ute to pro d uctivity, or (2) t he i nfor matio n co ntai ne d i n t he sig nal does n ot i ncrease ef ficie ncy. T his essay is mai nly a b o ut t he t he ory of sig nali ng a n d i n- for mation transfer in markets, and not mainly about hu man capital and labor markets. T here are ma ny w ho are more k no wledgeable about t he latter t ha n I. The paper concludes with t wo sections. The first of these discusses the u bi q uit o us us e of ti m e a n d t h e all o c ati o n of ti m e as a si g n al a n d as a s cr e e n- i ng device. T he seco n d foc uses o n t he pote ntially rat her large i nfor matio n pa- ra meter s hifts t hat t he i nter net may have caused a nd speculates a little o n ho w applied models of markets, organizations, and boundaries bet ween the m may have c ha nged a nd o n w hat is needed to capture t he effects of t hese pa- ra meter s hifts i n models of markets a nd t he eco no my.

T HE SI MPLEST J OB MARKET SIG NALI NG M ODEL T he i dea be hi n d t he job market sig nali ng mo del is t hat is t here are attrib utes of pote ntial e mployees t hat t he e mployer ca n not observe a nd t hat affect t he i ndividual’s subseque nt productivity a nd he nce value to t he e mployer o n t he job. Let us s u p pose t hat t here are j ust t wo gro u ps of peo ple. Gro u p 1 has pro- d uctivity or val ue to a ny e m ployer of 1, a n d Gro u p 2 has pro d uctivity of 2. I n t his exa m ple, t hese pro d uctivity val ues do not de pe n d o n t he level of i nvest- me nt i n t he sig nal. If t here were no way to disti nguis h bet wee n people i n these t wo groups then if both groups stay in the market, the average wage wo ul d be 2 – α , w here α is t he fractio n of t he populatio n i n group 1, a nd everyo ne would get t hat wage. If t he hig her productivity group t hroug h dis- satisfactio n or for a ny ot her reaso n, exits t his labor market, t he average pro- ductivity a nd t he wage drop to 1. T his p he no me no n w he n it occurs is so me- ti mes called t he adverse selectio n proble m, a label most co m mo nly applied to i ns ura nce markets. It is str uct urally t he sa me proble m t hat Akerlof describe d i n his fa mous paper o n used cars (le mo ns). 8

8 George Akerlof, “ The Market for Le mons: Qualitative Uncertainty and the Market Mechanis m,” Q uarterly Jo ur nal of Eco no mics , 1 9 7 1.

4 1 0 No w suppose t hat t here is so met hi ng called educatio n, w hic h we will de- n ot e by E, t h at c a n b e a c q uir e d or i nv est e d i n. It is ass u m e d t o b e visi bl e, a n d its ac q uisiti o n c osts differ f or t he t w o ty pes. Let us s u p p ose t hat t he c ost of E years of educatio n for group 1 is E, a nd t he cost for group 2 i ndividuals is E/2. For t his exa mple I a m goi ng to assu me t hat educatio n does not affect t h e i n divi d u al’s pr o d u ctivity. I d o t his p ur ely t o k e e p it si m pl e a n d n ot t o s u g- gest t hat hu ma n capital, i ncludi ng t hat acquired t hroug h educatio n, is so me- ho w not releva nt. I n later sectio ns, t he ass u m ptio n will be relaxe d. Equilibriu m i n a situatio n like t his, a nd i n ge neral, has t wo co mpo ne nts. First, give n t he ret ur ns t o a n d t he c osts of i nvesti ng i n e d ucati o n, i n divi d uals make ratio nal i nvest me nt c hoices wit h respect to educatio n. Seco nd, e mploy- ers have beliefs abo ut t he relatio n bet wee n t he sig nal a n d t he i n divi d ual’s u n- derlyi ng pro d uctivity. T hese beliefs are base d o n i nco mi ng data fro m t he mar- k et pl a c e. I n e q uili bri u m, t h e b eli efs m ust b e c o nsist e nt, t h at is, t h ey m ust not be disco n fir med by the inco ming data and the subsequent experience. There- fore, o ne co ul d say t hat t he beliefs m ust be acc ur ate . B ut o n e s h o ul d als o n o- tice t hat e mployers’ beliefs/expectatio ns deter mi ne t he wage offers t hat are ma de at vario us levels of e d ucatio n. T hese wage offers i n t ur n deter mi ne t he retur ns to i ndividuals fro m i nvest me nts i n educatio n, a nd fi nally, t hose re- tur ns deter mi ne t he i nvest me nt decisio ns t hat i ndividuals make wit h respect to education, and hence the actual relationship bet ween productivity and educatio n t hat is observed by e mployers i n t he marketplace. T his is a co m- plete circle. T herefore it is probably more acc urate to a say t hat i n eq uili- bri u m, t h e e m pl oy ers’ b eli efs ar e self-co n fir mi n g . T his m ay s o u n d li k e a mi n or restate me nt b ut it is i m porta nt. It is t he self-co n fir mi ng nat ure of t he beliefs t hat gives rise t o t he p ote ntial prese nce of m ulti ple e q uilibri a i n t he market.

I n t he exa mple at ha nd, suppose t hat group 1 i ndividuals set E 1 = 0 a n d i n- divi d uals i n gro u p 2 set E 2 = E*. Suppose further that e mployers, none of w ho m i ndividually i n flue nce i nvest me nt decisio ns by i ndividuals 9 , b eli ev e t h at if E < E * t h e n pr o d u ctivity is 1 a n d if E ≥ E *, t h e n pr o d u ctivity is 2. Wit h t h es e ass u m pti o ns, i n divi d u als i n gr o u p 1 will r ati o n ally s et E = 0 pr ovi d e d t h at 2 – E * < 1 Me mbers of gro u p 2 will ratio nally set E = E* provi de d t hat

T herefore t he c hoices will be ratio nal a nd t he expectatio ns co n fir med i n t he m ar k et if 1 < E * < 2. T ho ug h it is a hig hly stylize d exa m ple it has ma ny of t he ge neral pro perties of

9 I nvest me nt decisio ns are made by i ndividuals i n adva nce of k no wi ng wit h w ho m t hey will work a nd e mployers are suc h t hat no ne of t he m i ndividually i n flue nces materially t he perceived offers i n t he market place.

4 1 1 sig nali ng equilibria. T here is a co nti nuu m of equilibria, i n eac h of w hic h t h er e is m or e i nv est m e nt i n t h e si g n al t h a n t h er e w o ul d b e i n a w orl d of f ull i nfor matio n. Because i nvest me nt i n t he sig nal dissipates resources wit hout i m provi ng pro d uctivity, t he res ult is i nef ficie nt. Moreover t he eq uilibria are or derable by t he Pareto criterio n, t hat is, as yo u move fro m o ne eq uilibri u m to a not her, everyo ne is eit her worse off or no better off. T he sig nal act ually does disti ng uis h lo w- a n d hig h- pro d uctivity peo ple a n d t he reaso n it is able to do so is t hat t he cost of t he sig nal is negatively correlate d wit h t he u nsee n c haracteristic t hat is val ua ble t o e m pl oyers, i n t his case pr o d uctivity itself. T he eq uilibri u m is c haracterize d by a sc he d ule t hat gives t he ret ur ns to e d ucatio n, t hat is a wage for eac h level of e d ucatio n, a n d o pti mizi ng c hoices give n t hat s c h e d ul e by i n divi d u als i n t h e t w o gr o u ps. T h e e q uili bri u m is d e pi ct e d i n Fi g ur e 1. T h e w a g e s c h e d ul e is t h e d ar k li n e a nd ju mps fro m a wage level of 1 to a wage level of 2 at e*. E* is bet wee n 1 a nd 2 so t hat group 1 c hooses t he educatio n level 0 a nd group 2 c hooses E*. It is n ot ess e nti al t h at n et i n c o m e f or gr o u p 1 at e * is n e g ativ e as it is i n t his exa mple. We could make t he productivity levels for t he t wo groups 3 a nd 4. T he sa me sig nali ng res ult wo ul d occ ur wit h s uitable a dj ust me nts i n t he level of t he wage sc he d ule.

Fi g ure 1

T h er e ar e i n f a ct ot h er si g n ali n g e q uili bri a. F or e x a m pl e, t h er e c o ul d b e a mi ni m al l ev el of e d u c ati o n t h at gr o u p 1 i nv ests i n. B ut u nl ess it is a pr o d u c- tive i nvest me nt, t here is no reaso n to t hi nk t hat over ti me t he market would not discover t his a n d eli mi nate it as a n i nef ficie ncy t hat ca n be re move d cost- lessly. I n t he s pectr u m of eq uilibria describe d above, probably t he most i nte- resting is the most efficient one. That equilibriu m is the one in which E * = 1 + δ , w h er e δ is t o b e t h o u g ht of as a s m all p ositiv e n u m b er. I n t his e q ui- li bri u m,

4 1 2 w 1 = Ν 1 = 1 , a n d

w 2 = 2, a n d N 2 = 1. 5 – δ / 2, w here N i is i n c o m e n et of si g n ali n g c osts f or gr o u p i. O n t he ass u m ptio n t hat t he market will fi n d eq uilibria t hat are Pareto-ef fi- cie nt, o ne ca n ask w het her t here is a better equilibriu m i n t he Pareto se nse (everyo ne is better off) t ha n t he o ne described above. T he a ns wer is so me- ti mes yes. It i nvolves pooli ng a n d it de pe n ds o n t he relative sizes of t he t wo gr o u ps. W h at is r e ally g oi n g o n h er e is t h at t h e m ar k et is, i n a s e ns e, a cti n g as if it is tryi ng to maxi mize t he net i nco me of t he hig her pro d uctivity gro u p. I n so me cases t hat is acco mplis hed by recog nizi ng t hat it is too expe nsive for t hat gro u p to disti ng uis h itself. T he alter native is to a p pear u n differe ntiate d i n a pool. T hat clearly be ne fits group 1 as t hey are mistake n for t he average pr o d u ctivity, w hi c h is a b ov e 1. As a r e mi n d er, α is t he fracti o n of t he p o p ula- tio n i n gro u p 1. I n a pooli ng sit uatio n t he average pro d uctivity is 2 – α > 1. Gr o u p 1 w o ul d cl e arly pr ef er it. Gr o u p 2 pr ef ers it as w ell if 2 – α > 1. 5 – δ / 2, or

α < . 5 ( 1 – δ ) Si n c e δ c a n b e m a d e as s m all as w e w a nt, t h er e is a pr ef err e d p o oli n g e q uili b- riu m (i n t he Pareto se nse) if i n t his exa mple, group 1 is less t ha n half t he po p ulatio n. I n ge neral wit h discrete gro u ps, pooli ng wit h lo wer level gro u ps bec o mes attractive if t hey are relatively s mall, beca use t he hig her level gr o u ps do n’t give up too muc h by bei ng mistake n for t he average a nd t hey avoid t he si g n ali n g c osts. I n a l at er s e cti o n I will d es cri b e bri e fly a c as e i n w hi c h o n e c a n observe pooli ng a nd separati ng i n t he sa me equilibriu m. If you wanted to make the result in this market more efficient than the equilibria described above, you would tax education (assu ming that such a tax is not costly to i mpose and ad minister), making it more expensive for gro u p 2, a n d t hereby make it possible to lo wer t he level of e d ucatio n wit ho ut losi ng t he i nfor matio nal co nte nt of t he sig nal. T he reve nues ge nerated could be distrib ute d eq ually to all partici pa nts i n bot h gro u ps i n de pe n de nt of t heir educatio n c hoice. T he lu mp-su m co mpo ne nt of t he tax will t herefore not af- f e ct t h eir si g n ali n g b e h avi or. Let t be t he tax rate o n i nvest me nt i n e d ucatio n a n d let k be t he l u m p-s u m distrib utio n fro m t he tax reve n ues. T his distrib utio n goes to everyo ne. Let t he signaling level of education be E*. Group one individuals will rationally c h o os e n ot t o s e n d t h e si g n al if 2 – ( 1 + t) E * + k < 1 + k a n d gr o u p 2 will r ati o n ally s e n d t h e si g n al if 2 – (. 5 + t) E * + k > 1 + k . W e h av e a n e q uili bri u m t h e n if

4 1 3 .

W e will pi c k t h e ef fi ci e nt e n d of t h e si g nli n g s p e ctr u m by s etti n g

T he l u m p s u m distrib utio n is eq ual to t he tax reve n ues so t hat T hus t he equilibriu m net i nco mes are

a n d

As δ beco mes s mall and t beco mes very large, k approaches (1 – α ), a n d hence both net inco mes approach (2 – α ). T his, t h e r e a d er will r e c all, is t h e pooli ng equilibriu m outco me i n ter ms of net i nco me. Here ho wever, t here is n o u p p er li mit o n t h e siz e of gr o u p 1, n a m ely α . Si g n ali n g c osts (s o ci ally ) ar e negligible as E* a p proac hes zero, t ho ug h t he private margi nal costs of sig nal- i ng (i ncludi ng t he tax) approac h 1. To su m marize, o ne ca n get rid of t he i nef ficie ncy a nd keep t he sig nali ng a nd t he i nfor matio nal co nte nt of t he sig nals wit h a n appropriate tax o n t he sig nali ng activity. T he effect is t o re distri b ute i nc o me i n s uc h a way as t o re- pro d uce t he res ults of t he pooli ng eq uilibri u m. It co ul d be t hat t he i nfor ma- ti o n t he sig nal carries is itself pr o d uctive. I n t hat case it is i m p orta nt t hat t he e q uili bri u m a ct u ally b e a s e p ar ati n g o n e. 1 0 T hat is t he case here. Se parati ng is retai ne d i n t his eq uilibri u m. Gro u ps are correctly i de nti fie d. T he use of a tax o n t he sig nali ng activity re d uces t he level of t he sig nal re q uire d t o disti ng uis h t he groups, a nd he nce reduces t he i nef ficie ncy of t he u ntaxed sig nali ng equi- li bri u m.

THE T WO-GROUP MODEL WHEN EDUCATION ENHANCES HU MAN CAPITAL A N D ALS O SERVES AS A P OTE NTIAL SI G NAL It will nat urally occ ur t o t he rea der t o ask w hat is t he effect o n t he market sig- nali ng t hat we have j ust exa mi ne d, at least i n t he co ntext of labor markets, w he n t he sig nal also co ntrib utes directly to pro d uctivity for t he i n divi d ual w or ker. 1 1 T his sectio n addresses t he subject by modifyi ng t he t wo-group mo- del of t he prece di ng sectio n to allo w e d ucatio n to be pro d uctive. I will make

1 0 I nf or m ati o n is dir e ctly pr o d u ctiv e, f or e x a m pl e, if t h er e is a j o b all o c ati o n or tr ai ni n g d e cisi o n t hat is ma de more ef ficie ntly wit h acc urate k no wle dge of t he ty pe. I n t hat case t he pooli ng eq uili- bri u m w o ul d b e i n ef fi ci e nt. 1 1 Act ually t his iss ue d oes n ot j ust arise i n la b or markets. Ge nerally sig nals ca n c ha nge t he val ue of t he pro d uct. For exa m ple, i n t he use d car case, a warra ntee c ha nges t he val ue of t he b u n dle, t he car a n d t he warra ntee i n a d ditio n to carryi ng i nfor matio n abo ut t he pro d uct itself.

4 1 4 t he mo del slig htly more ge neral i n ter ms of f u nctio nal for ms. Let s i( E) be t he val ue of a worker of ty pe I wit h e d ucatio n E to a n e m ployer. Here I = 1,2 a n d we ass u me t hat s2 ( E ) > s1 ( E ), a n d s2 ( E ) > s1 ( E ). L et c i( E) be t he c ost of i n- vesti ng i n E u nits of e d ucatio n for gro u p I a n d ass u me t hat c1 ( e) > c2 ( e), a n d t h at c1 ( E ) > c2 ( E ). T hese assu mptio ns si mply capture t he idea t hat t he group wit h t he hig her pro d uctivity also has t he lo wer costs of sig nali ng. As before, t h e i n divi d u al’s ty p e is n ot dir e ctly o bs er v a bl e. W e will als o ass u m e t h at S i( E ) is c o ncave, t hat C i( E ) is co nvex a n d he nce t hat t he net i nco me f u nctio n N i( E ) = S i( E ) – C i( E) is c o ncave. T here are t hree q ualitatively differe nt ki n ds of eq uilibria i n t his market. T he first is a f ully ef ficie nt se parati ng e q uili bri u m. It is easiest t o see t he ty pes of eq uilibri u m pictorially. I n Fig ure 2, we have vario us for ms of net i nco me pl ott e d as a f u n cti o n of E. L et V 1 ( E ) = S 2 ( E ) – C 1 ( E ). O ur i nt er est i n t h e f u n c- tio n V clearly relates to t he q uestio n of w het her gro u p 1 will a do pt t he gro u p 2 sig nal a nd he nce create a proble m wit h t he separati ng equilibriu m. I n Fi g ur e 2, y o u will s e e t h e f u n cti o ns N , N , a n d V. T h e p oi nts E * a n d E * 1 2 – 1 2 are t he poi nts t hat maxi mize N 1 a n d N 2 . T h e p oi nt E is t he largest val ue of E s uc h t hat V is larger t ha n N *, t h e m a xi m u m v al u e of N . I n Fi g ur e 1, y o u c a n 1 – 1 see t hat E * is t o t he rig ht of E . T his mea ns t hat as lo ng as t he wage offer 2 – schedule ju mps up fro m s 1 ( E ) t o s 2 ( E ) t o t h e ri g ht of E , i n divi d u als i n gr o u p 1 will h av e n o i n c e ntiv e t o i mit at e t h e si g n ali n g b e h avi or of gr o u p 2. T his is t he f ully ef ficie nt se parati ng e q uili bri u m. T he sig nal carries acc urate

Fi g ure 2

4 1 5 infor mation, and the invest ments in education are the efficient ones. The outco me is as if there was perfect infor mation in the market place. In esse nce, t he t wo groups are suf ficie ntly differe nt i n ter ms of so me co mbi na- tio n of t he productivity of educatio n as hu ma n capital a nd t he costs of edu- catio n t hat t he fully ef ficie nt outco me survives as a n equilibriu m. Note ho w- ever, t hat t he negative relatio n bet wee n pro d uctivity a n d costs really matters.

If sig nali ng costs were t he sa me, t he n t he fu nctio ns N 2 and V would be the s a m e a n d w e c o ul d n ot h av e t his m ar k et o ut c o m e. W h e n si g n ali n g c osts ar e the sa me one can never prevent group 1 fro m i mitating group 2 behavior w h e n it is t o t h eir a dv a nt a g e t o d o s o. No w t ur n to Fig ure 3. It is esse ntially t he sa me pict ure as Fig ure 2 exce pt – t h at E 2 * is t o t h e l eft of E . T his mea ns t hat if t he wage sc he d ule j u m ps to s 2 ( E ) bef ore E 2 *, t he group 1 will i mitate group 2 be havior a nd dis ma ntle t he se- parati ng eq uilibri u m. To preve nt t his we will s pecify t hat t he wage sc he d ule – ju mps up to s 2 ( E ) at E + δ . S u bj e ct t o a q u ali fi c ati o n t h at I will c o m e t o i n a mi n ute, t his is a se parati ng eq uilibri u m. To ac hieve t he se paratio n, t he gro u p 2 i nvest me nt i n e d ucatio n is p us he d u p above t he ef ficie nt level, i n or der for t h e i n divi d u als i n t h at gr o u p t o a c hi ev e t h e si g n ali n g eff e ct a n d av oi d i mit a- tio n by group 1. So here we have t he sa me ki nd of result t hat we had i n t he no- hu ma n-capital case. Here t he retur n to i nvesti ng i n educatio n has a sig- naling co mponent and a hu man capital co mponent. The for mer can push t he levels of i nvest me nt above t he full i nfor matio n opti mu m. I n t he first mo- del t he h u ma n-ca pital effect was zer o, leavi ng o nly t he sig nali ng effect, a n as-

Fi g ure 3

4 1 6 su mptio n t hat e nsures t hat overi nvest me nt will occur i n a ny separati ng equi- li bri u m i n w hi c h si g n ali n g o c c urs. There re main t wo issues. One is whether and under what circu mstances, pooli ng will destroy t he hypot hesized separati ng equilibriu m a nd t he seco nd is, “ Ca n o ne tax t he sig nal i n suc h a way as to i mprove market perfor ma nce?” P o oli ng first. Let α be t he fractio n of t he total po p ulatio n i n gro u p 1. I n a p o oli n g e q uili bri u m, t h e w a g e f or all will b e α s1 ( E ) + ( 1 – α ) s2 ( E ) f or a ny l e- vel of E t hat e merges. T his ca n not break a fully ef ficie nt separati ng equili- briu m because group 2 ca n not be better off. T hus if pooli ng breaks a sepa- rating equilibriu m it can only be in the case where the more productive group has bee n forced, for sig nali ng purposes, above its opti mal level of E fro m t he sta ndpoi nt of i nvest me nt i n hu ma n capital. Let W ( E ) = α s1 ( E ) + ( 1 – α ) s ( E ). By m a ki n g α ar bitr arily s m all w e c a n m a k e W as cl os e t o s ( E ) as w e 2 – 2 wa nt. No w let us look agai n at Fig ure 2. If yo u start at E and reduce E and m ov e t o t h e l eft a n d u p t h e c ur v es N 2 a n d V, it is cl e ar t h at b ot h gr o u ps ar e b ett er off. F or s m all α , t he net i nco me fu nctio ns W-c 1 a n d W-c 2 ar e cl os e t o V and N 2 . He nce for a ra nge of α at t he lo w e n d, pooli ng will break t he se pa- rati ng o utco me. For t hese cases, t he nat ural c hoices of eq uilibri u m levels of E are t hose t hat lie bet wee n t he e d ucatio n levels t hat maxi mize W-c 1 a n d W-c 2 . As α rises, eve nt ually t he f u ncti o n W-c will f all b el o w t h e l ev el of gr o u p 2 – 2 net i nco me at t he se parati ng poi nt E , at w hi c h p oi nt t h e p o oli n g o ut c o m e is not capable of breaking the separating equilibriu m. To su m marize, in the case of a n i nef ficie nt separati ng equilibriu m, t here exist pooli ng equilibria t hat are Pareto-s u perior to t he se parati ng eq uilibri u m provi de d t he size of t he lo wer productivity group is belo w so me t hres hold level. If t he market has a w ay of fi n di n g t h e p o oli n g e q uili bri a, t h ey will br e a k t h e s e p ar ati n g o n e. We tur n no w to t he proble m of i mprovi ng t he ef ficie ncy of t he market. We sa w i n t he prece di ng sectio n t hat i n t he case w here t here is no h u ma n ca pital co m po ne nt of pro d uctivity, o ne ca n i n pri nci pal co me arbitrarily close to t he ef ficie nt outco me t hroug h appropriate taxes. T he sa me is true w he n t he sig- n al a d ds t o t h e i n divi d u al’s pr o d u ctivity as w ell as f u n cti o ni n g as a si g n al. W e have alrea dy see n t hat i n so me cases t he market eq uilibri u m itself is ef ficie nt i n t he se nse of maxi mizi ng total net i nco me as part of a se parati ng eq uilibri- u m. We no w want to sho w that one can achieve the sa me result in the case w here t he hig her productivity group has bee n forced by sig nali ng co nsidera- tio ns to move above its ef ficie nt level of i nvest me nt i n h u ma n ca pital. To ac hieve a n ef ficie nt outco me, we wa nt to maxi mize total net i nco me. To do t hat we nee d to have si = ci f or i = 1, 2. T o a c c o m plis h t h at if it is p ossi bl e, we need a wage fu nctio n t hat i nduces t hese c hoices. T he wage fu nctio n ca n be thought of as a schedule of taxes on education superi mposed on a wage f u nctio n t hat sets wages eq ual to pro d uctivity. Si nce eac h i n divi d ual i n eac h gro u p co nsi ders all possible levels of E i n maki ng its e d ucatio n i nvest me nt de- cisi o n, it t ur ns o ut t o b e e asi er t o s olv e t his pr o bl e m i n dir e ctly r at h er t h a n di- rectly. Let us s u p pose t herefore t hat co u nter to fact, t here is a co nti n u u m of ty pes s peci fie d by t he para meter z, w here pro d uctivity for ty pe z is zs 1 +( 1-z)s 2 a n d c osts f or ty pe z are zc 1 +( 1-z)c 2 . Let E(z) maxi mize

4 1 7 z( s1 – c1 ) + ( 1 – z) ( s2 – c 2 ). N ote t hat E(z) decli nes as a f u ncti o n of z. I nvert E(z) t o get Z( E). If t he wage s c h e d ul e w e ar e l o o ki n g f or is w ( e ), t h e n i n divi d u als will h av e m a xi miz e d by s etti n g

w = zc 1 + ( 1 – z) c2 . Substitute Z( E) i nto t he differe ntial equatio n above a nd i ntegrate wit h re- s pect to E, setti ng t he co nsta nt so t hat total wages eq ual total o ut p ut. T his wage schedule and/or the i mplied tax schedule will induce the efficient c h oi c es of E. 1 2 T he reader notices t hat t he opti mal wage sc hedule above does not depe nd o n t he distrib utio n of z i n t he po p ulatio n. 1 3 T herefore let us ass u me t hat al- m ost all t h e w ei g ht i n t h e distri b uti o n is at eit h er z = 0 or z = 1. T his is t h e t w o gro u p case we are st u dyi ng. To su m marize, with t wo groups and education as productive hu man capi- tal, o ne ca n have a sig nali ng equilibriu m wit h full ef ficie ncy or overi nvest- me nt i n educatio n by t he more productive group. You ca n also have a pooli ng eq uilibri u m t hat do mi nates t he se parati ng eq uilibri u m provi de d t hat t he less- productive group is not too large. And finally there exists a tax/subsidy sc he me t hat produces a fully ef ficie nt separati ng outco me as a n equilibriu m.

A M ODEL WIT H SIG NALI NG, SELECTI O N A ND P O OLI NG 1 4 T he m o del we are g oi ng t o l o ok at brie fly i n t his secti o n is of i nterest f or t w o reas o ns. First it ill ustrates a case i n w hic h t here is b ot h p o oli ng a n d se parati ng co m po ne nts of t he eq uilibri u m, a n d seco n d, it s ho ws t hat t he critical criteri- o n for havi ng a se parati ng co m po ne nt i n a n eq uilibri u m is t hat t he net be ne- fits of iss ui ng t he sig nal are p ositively c orrelate d wit h a n u nsee n attri b ute t hat c o ntri b utes p ositively t o pr o d uctivity. T his p ositive c orrelati o n ca n res ult as i n prece di ng exa m ples, fro m sig nali ng costs t hat are negatively correlate d wit h t he val ue d attrib ute. T hat is a s uf ficie nt b ut not a necessary co n ditio n for a se parati ng eq uilibri u m. I n t his very nice exa m ple t he i nfor matio n is acq uire d at a fixed cost a nd t he positive correlatio n co mes fro m subseque nt discovery of t he attrib ute post-e m ploy me nt. T h e i d e a b e hi n d t h e m o d el is t h at t h e v al u e of i n divi d u als t o fir ms is n ot di- r e ctly o bs er v e d, at l e ast at t h e ti m e of hiri n g. T h e v al u e w hi c h w e will d e n ot e by q is distri b ut e d o n t h e i nt er v al [ q mi n , q m a x ]. L et t h e distri b uti o n of q i n t h e

1 2 N ote t hat w – [ zc 1 + ( 1 – z) c2 ] = z ( E ) [ c1 – c2 ) < 0, so t hat t he seco n d or der co n ditio n for a m a xi m u m is s atis fi e d. T h e o pti m al t a x s c h e d ul e will a ct u ally c o nsist of a risi n g t a x o n e d u c ati o n at all levels of e d ucatio n co mbi ne d wit h a l u m p s u m distrib utio n to everyo ne so t hat net tax re- v e n u es ar e z er o. I n t his e x a m pl e, gr o u p 1 will r e c eiv e a n et s u bsi dy a n d gr o u p 2 will p ay a n et t a x. 1 3 T his is a g e n er al pr o p erty of t h e c o nti n u o us c as e, w h e n t h e o bj e ctiv e is, as it is h er e, t o m a xi- mize net i nco me by i nduci ng t he ef ficie nt levels of i nvest me nt i n hu ma n capital. 1 4 T his exa m ple was develo pe d by Professor E d war d Lazear. It ill ustrates nicely t he coexiste nce of pooli ng a n d se parati ng i n eq uilibri u m a n d also a differe nt str uct ural co n ditio n t hat per mits t he sig nali ng a n d selectio n to occ ur. It ca n be fo u n d i n his book, E d war d Lazear, Perso n nel Eco no mics .

4 1 8 po p ulatio n be f(q). I n divi d uals have a c hoice. T hey ca n work for fir ms t hat do not distinguish a mong the m and hence pay everyone the sa me a mount. Or t hey ca n work for fir ms t hat i ncur a n expe nse, de noted by e, as a result of w hic h, t he fir m eve nt ually lear ns t he val ue of q for t he i n divi d ual a n d pays t he perso n accor di ngly. I n eq uilibri u m t his cost, w hic h is t he sa me for everyo ne, is passed o n to t he i ndividual i n t he for m of a reductio n i n co mpe nsatio n. Labor markets are assu med to be co mpetitive. For those individuals who c hoose t he fir ms w here t hey are disti nguis hed, t he wage is q-e. If t he i ndi- vidual c hooses a fir m t hat does not i ncur t he cost a nd does not disti nguis h a mo ng workers, t he n t he co mpe nsatio n is t he average value of t he i ndividuals w ho work for t hat fir m. Let us s u p pose t hat t he average val ue of t he workers i n t he pooli ng fir ms is q– . If w e c o nsi d er t h e o pti mizi n g d e cisi o ns of i n divi d u als, it is cl e ar t h at if q – e > q– t he n t he i ndividual will c hoose to work for t he separati ng fir m a nd co nverse- ly. Let q* = q– + e. I n e q uili bri u m, q– is t he act ual average val ue of t hose w ho w or k i n t h e p o oli n g fir ms, s o t h at

w here F(q) is t he cu mulative distributio n fu nctio n for t he attribute q. T he e q uili bri u m is ill ustr at e d i n Fi g ur e 4.

Fi g ure 4

4 1 9 T h os e wit h q ≤ q * r e c eiv e q– a n d ar e p o ol e d, w hil e t h os e wit h q > q * r e c eiv e q - e a n d ar e i d e nti fi e d a n d ar e i n t h e s e p ar ati n g p art of t h e e q uili bri u m. It is possible t hat everyo ne will be poole d. T his occ urs w he n e, t he cost of disco- very, is large e n o ug h t hat q m a x – e is less t ha n t he u nco n ditio nal mea n of t he w h ol e distri b uti o n. A n ot h er vi e w of t h e e q uili bri u m is pr ovi d e d i n Fi g ur e 5. Here we plot t he f u nctio ns q*, a n d E ( q |q ≤ q * ) + e, as a f u n cti o n of q *. T h e seco n d f u nctio n is u p war d-slo pi ng a n d will cross t he 45- degree li ne, u nless e is large e noug h to keep t he seco nd fu nctio n above t he 45-degree li ne for all q *. T h e cr oss ov er p oi nt is t h e e q uili bri u m l ev el of q *. 1 5 T h e si g n al is t h e c h oi c e of w hi c h ty p e of fir m t o g o t o a n d it is p arti ally, b ut n ot c o m pl et ely, i nf or m ativ e a b o ut t h e i n divi d u al’s v al u e t o t h e fir m. I n ot e d i n t he first mo del t hat t he sig nal retai ns its i nfor matio nal co nte nt i n eq uilib- ri u m if t he cost of t he sig nal is negatively correlate d wit h t he ( har d-to-ob- serve) val ue d attri b ute. I n t his case, t he c ost of t he sig nal is a c o nsta nt acr oss people but because of t he subseque nt discovery a nd adjust me nt, t he net be- ne fit to t he i n divi d ual of iss ui ng t he sig nal is positively correlate d wit h t he val ue d attri b ute. It is p ossi ble t hat t he i nf or mati o n carrie d by t he sig nal i ncreases ef ficie ncy. T hat circu msta nce i ncreases t he retur n to goi ng i nto t he separati ng side of

Fi g ure 5

1 5 W hile t he c o n diti o nal ex pecte d val ue f u ncti o n is u p war d-sl o pi ng, its sl o pe is n ot necessarily less t ha n o ne, a nd depe nds o n f(q). It is t herefore i n pri ncipal possible to have a n odd nu mber of m ulti ple crossovers. If t here are t hree, t he mi d dle o ne will be u nstable a n d t he o nes at t he o ut- si de will cross over wit h a slo pe of less t ha n o ne a n d be stable. To see t his, note t hat if at so me p oi nt of ti m e, q * – e > q– , t h e n q * will ris e, a n d c o nv ers ely.

4 2 0 t he market a n d has t he overall effect of i ncreasi ng t he size of t he gro u p t hat se n ds t he sig nal. I s ho ul d also note t hat it has bee n i m plicitly ass u me d here t hat t he i nfor matio n abo ut pro d uctivity, o nce it is acq uire d, is p ublic. T hat would force t he e mployer to pay t he e mployee his or her productivity mi nus t he disc overy c ost. If, o n t he ot her ha n d, t he disc overy is private, t here will be a negotiation bet ween the e mployer and the e mployee over the net inco me, a nd t hat will have t he effect of lo weri ng t he net i nco me of t he e mployee re- lative to t he case i n w hic h t he i nfor matio n is p ublic. T h us o ne s ho ul d ex pect t he size of t he pooli ng co mpo ne nt of t he market equilibriu m to rise w he n t he dis c ov er y is priv at e.

T HE GENERAL C ONTINU OUS M ODEL 1 6 I wo ul d no w like to establis h so me reaso nably ge neral pro perties of sig nali ng m o d els. 1 7 Follo wi ng t hat we will look at i mprovi ng t he equilibriu m perfor- ma nce of t he market t hroug h taxes a nd subsidies, usi ng t he approac h of op- ti mal taxatio n wit h i mperfect i nfor matio n. T he variables are n, sta ndi ng for so me attribute t hat is (a) not directly observable, a nd (b) valuable to e m- ployers, a n d y, w hic h is years of e d ucatio n. T he latter is observable a n d may be valuable to e mployers. T he fu nctio ns wit h need are S( n,y) w hic h speci fies a n i ndividual’s productivity or value to a n e mployer as a fu nctio n of educa- tio n a n d n, w hic h I will he nce fort h refer to as ability. T he re mai ni ng f u nc- tio ns are c(y, n) w hic h s peci fies t he cost of e d ucatio n as a f u nctio n of t he sa me t wo variables a nd w(y) w hic h is t he wage offered to a n i ndividual w ho pre- se nts hi m or herself i n t he market wit h e d ucatio n of y. Si nce t his is q uite fa- mili ar t errit or y, I will pr o c e e d f airly q ui c kly. T h e e q uili bri u m is d e fi n e d by t w o co nditio ns. First, give n w(y), i ndividuals maxi mize i nco me net of educatio n costs wit h res pect to y. T h us t hey maxi mize w(y) – c(y, n) by setti ng

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

T his hol ds for all n. T he seco n d or der co n ditio n, w ( y) – cyy ( y, n )< 0 m ust h ol d. The second condition is that e mployers’ experience in the market over ti me m ust be co nsiste nt wit h t he offers t hey are maki ng, so t hat for all n ( y) = s( y, n ). Ignore the second-order conditions for the mo ment. Since by assu mption

1 6 T his sectio n is so me w hat tec h nical. It is not dif ficult mat he matics but does require a ge neral k n o wle dge of differe ntial e q uati o ns a n d t he calc ul us of variati o ns. It is n ot esse ntial f or t h ose i n- terested i n lear ni ng t he basic properties of sig nali ng equilibria a nd ca n be skipped by readers w ho are willi ng to acce pt t hat t he pro perties cite d i n t he earlier exa m ples s urvive i n t he ge neral case. T he seco n d best o pti mizi ng calc ulatio ns are of so me i nterest i n i nter preti ng w hat is goi ng o n i n t h e m ar k et. 1 7 Muc h of t he material i n t his sectio n is take n fro m parts of t he t he paper, “ Co mpetitive a nd Opti mal Respo nses to Sig nals: A n A nalysis of Ef ficie ncy a nd Distributio n,” by Mic hael Spe nce, Jo ur nal of Eco no mic Theory, 1 9 7 4.

4 2 1 sn > 0, o n e c a n i n pri n ci pl e s olv e t h e e q u ati o n a b ov e f or n i n t er ms of w a n d y, say n= N( w,y). S ubstit uti ng i n t he first or der co n ditio ns, we have

( 1 ) ( y) = cy( y, N ( , y) ). T his is a first-or der, or di nary differe ntial eq uatio n. It has a o ne- para meter fa mily of sol utio ns t hat do not cross eac h ot her. I n pri nci ple eac h me mber of t his o ne-para meter fa mily ca n be part of a market-sig nali ng equilibriu m. 1 8

If c ( 0, n ) = 0 f or all n a n d if c y n < 0, t h e n ass u mi n g c y > 0, c (y, n ) will b e d e- clining in n and that co mbined with an up ward sloping w(y) ( without which n o o ne w o ul d i nvest i n e d ucati o n), e ns ures t hat y is a n i ncreasi ng or at least a no n-decreasi ng fu nctio n of n. Let t he net i nco me fu nctio n be N ( y, n ) = w ( y) – c( y, n) .

T his f u nctio n has t he pro perty t hat N y is a n i ncreasi ng f u nctio n of n, w hic h e ns ures t hat if N y = 0 f or a p arti c ul ar v al u e of n, t h e n if y o u r ais e n, N y will b e greater t ha n zero a n d t he maxi m u m will occ ur at a hig her val ue of y. Differe ntiati ng (1) wit h res pect t o y we have

, w hic h mea ns t hat t he seco n d or der co n ditio n is i n fact satis fie d. Wit ho ut bei ng too for mal abo ut it, e d ucatio n or a ny pote ntial sig nal tra ns- mits i nfor matio n i n eq uilibri u m if its costs absol utely a n d at t he margi n de- cli ne as t he u nsee n val ue d attrib ute i ncreases. T his is a s uf ficie nt co n ditio n. I n a l at er s e cti o n I will s h o w by e x a m pl e t h at it is n ot a n e c ess ar y c o n diti o n. Si n c e f or ev er y l ev el of n, i n e q uili bri u m w ( y) s( n , y), diff er e nti ati n g w e hav e

beca use s n > 0 a n d > 0. T his i m pli es t h at i n e q uili bri u m, t h e priv at e r et ur n to e d ucatio n is hig her t ha n its direct co ntrib utio n to pro d uctivity, beca use of t he seco nd ter m i n t he equatio n above. T his seco nd ter m is t he sig nali ng ef- fect. It is t he part of t he private ret ur n t o t he i nvest me nt i n e d ucati o n t hat is tie d to t he u nobserve d level of n. Si nce w = c y, t he above i neq uality i m plies t h at sy – cy < 0, s o t h at i n e q uili bri u m f or all n, t h e i nv est m e nt i n e d u c ati o n is hig her t ha n it wo ul d be wit h perfect i nfor matio n. If n were observable, t he n i ndividuals would be paid s( n,y), a nd t hey would select educatio n to maxi- miz e s – c by s etti n g sy = cy. N ot e t h at if sn 0 f or all n, t h e n t h e si g n ali n g ef- fect goes a way. T he attrib ute n is still u nobservable to e m ployers b ut i n divi- duals make ef ficie nt educatio n i nvest me nt decisio ns as t hey have t he i nfor-

1 8 T he sol utio ns ca n not cross beca use if t hey di d, t he n t he derivative at a give n poi nt i n ( w,y) space would have t wo values, w hic h co ntradicts t he state me nt t hat w has a well-de fi ned value F( w,y) at every poi nt. As a co nseq ue nce, let us ass u me t hat t he fa mily of sol utio ns is w(y, K) w here

K is a para meter. If w K > 0 a ny w here, t he n it is true every w here, si nce to assu me t he opposite wo ul d mea n t hat t he sol utio ns cross.

4 2 2 matio n t hey nee d to make t he i nvest me nt decisio n o pti mally, a n d t hey are t he o nly o nes w ho need t hat i nfor matio n. As noted earlier, there is a one-para meter fa mily of equilibriu m wage sc hedules t hat do not cross. Let t he para meter be k a nd de note t he sc hedules by w (y, k ). As t h e s ol uti o ns d o n ot cr oss, wit h o ut l oss of g e n er ality w e c a n as- su me t hat w k > 0. Net i nco me to t he i n divi d ual is N = w – c. Differe ntiati ng p arti ally wit h r es p e ct t o k, wit h n h el d c o nst a nt, w e h av e

( 2 ) N k = w k + ( w y – cy) ( ∂ y |∂ k ) = w k > 0. T hus a s hift fro m o ne equilibriu m to a not her makes everyo ne better or worse off toget her. T he eq uilibria are or derable by t he Pareto criterio n. W hat ha p- pe ns as o ne moves fro m o ne eq uilibri u m to a not her is t hat t he exte nt of over- i nvest me nt i n t he sig nal is i ncreasi ng or decli ni ng for everyo ne. I have to co n- fess t hat at t he ti me, eve n after discoveri ng t hat t here mig ht be multiple equilibria, I did not expect t hat t here would be suc h a si mple relatio ns hip a mong the m in ter ms of market perfor mance. We ca n use (2) a b ove a n d t he fact t hat w s t o derive t he effect of a s hift i n t he equilibriu m o n t he levels of i nvest me nt i n educatio n. N = w – c = s – c. Differe ntiati ng wit h res pect to k, hol di ng n co nsta nt a n d usi ng (2) we have

.

Fi n ally, fr o m t h e e q uili bri u m c o n diti o n w = cy , differe ntiati ng wit h res pect t o k wit h n fi x e d, w e h av e

.

T his gives us a pretty co mplete picture of w hat happe ns w he n t he equilibriu m wage sc he d ule s hifts u p. It beco mes flatter, i n d uci ng lo wer levels of i nvest- ment in education. Average wages decline and average education costs de- cli ne more, so t hat net i nco mes rise a n d everyo ne is better off. Fig ure 6 re- prese nts t he fa mily of equilibriu m wage sc hedules. As t hey rise t hey beco me flatter a n d t he ra nge of i nvest me nt i n e d ucatio n s hifts to t he left. Fig ure 6 s ho ws t he ra nge of i nvest me nt i n educatio n for eac h of t he sc hedules. As t he sc hedules rise a nd beco me flatter, t he ra nges of i nvest me nt i n edu- c ati o n m ov e t o t h e l eft, t h at is t h ey f all. W hile it is per ha ps i nteresti ng t o n ote t hat t he sig nali ng effect ca uses over- i nvest me nt i n t he sig nal by t he sta n dar d of a worl d i n w hic h t here is perfect i nf or m ati o n, t h at is n’t t h e w orl d w e liv e i n. T h us it m ay b e m or e i nt er esti n g t o exa mi ne t he equilibriu m outco mes i n co mpariso n to various seco nd-best out- co mes that ackno wledge that there is an infor mational gap and asy m metry t hat ca n not si mply be re moved by a wave of t he pe n. I will b e gi n by l o o ki n g at t h e m a xi miz ati o n of t ot al n et i n c o m e. H er e as w e s hall see, o ne ca n ac hieve t he ef ficie nt levels of i nvest me nt i n e d ucatio n for all n. T he req uire d tax i n effect u n does t he sig nali ng effect. T he reaso n t hat t his is a cl e a n c as e is b e c a us e a d oll ar of n et i n c o m e is of e q u al v al u e r e g ar d-

4 2 3 Fi g ure 6 less of t he reci pie nt, we d o n ot get i nt o t he b usi ness of tra di ng off i nc o me di- stri b uti o n agai nst t he o bjective of ef ficie ncy. F oll o wi ng t his brief a nalysis, we will look at social welfare f u nctio ns t hat are not li near i n net i nco me. For t hese, t here will be a tra deoff bet wee n ef ficie ncy a n d distrib utio n. I will s ho w that the market outco me is more akin to the outco me of maxi mizing a con- vex (t hat is t o say a nti-egalitaria n) s ocial welfare f u ncti o n. T his is n ot a s ur- prisi ng result si nce t he fu nctio n of sig nali ng is to disti nguis h lo w- fro m hig her- pro d uctivity i n divi d uals, w hic h res ults i n a move a way fro m egalitaria n o utco mes. I do also wa nt to e m p hasize at t he o utset t hat t he p ur pose of t his a nalysis is not to pro pose or pro mote taxi ng or s ubsi dizi ng sig nals b ut rat her to help s hed additio nal lig ht o n t he properties of t he co mpetitive equili- bri u m.

MAXI MIZI NG T OTAL NET I NC O ME We will ass u me t hat t he u nobservable attrib ute n is distrib ute d i n t he po p ula- tio n accordi ng to t he de nsity fu nctio n f( n), a nd t he cu mulative distributio n will be re prese nte d by F( n). T he total net i nco me t he n is

( 3 ) .

H er e N ( y, n ) = w ( y) – c( y, n ) as before. T he co nstrai nts are t wo. I n divi d uals c h o ose rati o nally s o t hat w = cy a n d total gross wages have to eq ual total pro- d uctivity or val ue ge nerate d to e m ployers:

4 2 4 ( 4 ) .

O ur goal here is to select a sc he d ule w(y) t hat ca uses e d ucatio n c hoices y( n) for eac h n, t hat maxi mize total net i nco me, net t hat is of e d ucatio n costs. T his is act ually pretty straig htfor war d. Usi ng (4), we ca n re place w(y) i n (3) a n d choose y(n) to maxi mize

.

T his is t h e si m pl est f or m of a c al c ul us of v ari ati o ns pr o bl e m. 1 9 T h e s ol uti o n is a sc he d ule y( n) t hat ca uses sy = cy , f or all l ev els of n. T h e r e a d er will n ot e t h at t he i mplied levels of i nvest me nt i n educatio n are t hose w hic h would occur if n were observable, t hat is i n t he hy pot hetical worl d of perfect i nfor matio n. T hey are t heref ore als o as a res ult t he ef ficie nt levels of i nvest me nt i n e d uca- tio n for eac h n: e d ucatio n is i nveste d i n u p to t he poi nt w here at t he margi n its direct co ntrib utio n to pro d uctivity (t he h u ma n ca pital effect) is eq ual to t he margi nal cost. T hat is to say, to maxi mize net i nco me, t he wage sc hedule is set s o as t o re m ove t he sig nali ng effect, t h o ug h e d ucati o n still carries t he i n- for matio n a n d acts as a sig nali ng. T he private part of t he ret ur n to t he sig nal ( w hic h has to do wit h disti nguis hi ng o ne i ndividual fro m a not her a nd he nce falls into the zero-su m/redistribution aspect of the market mechanis m) is si mply re moved by the opti mal tax or wage schedule. To find the required wage sc he d ule, yo u i nvert t he o pti mal sc he d ule y( n) to r(y), s ubstit ute t hat i n t h e o pti mizi n g c o n diti o n w ( y) = cy( y, r ( y)), a n d i ntegrate t o get

( 5 ) .

T he para meter w(0) is t he n set to cause t he breakeve n co nditio n (4) to be m et. I n divi d u als ar e li k ely b ei n g p ai d t h eir pr o d u ctivity i n t h e m ar k et pl a c e, s(y,r(y)). T herefore o ne ca n get to t he desire d res ult by i m posi ng a tax /s ub- si dy o n e d ucatio n of t(y) = s(y,r(y)) - w(y) w here w(y) is de fi ne d by (5) above. If y o u differe ntiate t(y) wit h res pect t o y y o u have

t ( y) = sy + sn r – w = sy – cy + sn r = sn r . T he last ter m i n t he e q uati o n is t he sig nali ng effect at t he margi n of a c ha nge i n e d ucatio n. T his t he n says t hat t he o pti mal tax at t he margi n is eq ual to t he sig nali ng effect. It takes t hat part of t he private ret ur n t o e d ucati o n a way, leav- i ng o nly t he h u ma n ca pital effect or t he direct co ntrib utio n to pro d uctivity. O ne ca n also ac hieve t he sa me result wit h a n i nco me tax. Let i nco me be I. Its relati o n t o e d ucati o n is give n by I = s(y,r(y)). I nvert t hat f u ncti o n t o get

1 9 A goo d treat me nt of t he calc ul us of variatio ns ca n be fo u n d i n Co ura nt. Yo u dis place y by δ θ ( n) , differe ntiate wit h res pect to δ a nd i nsist t hat t he result be equal to zero for all θ ( n) . T his giv es us T h at i n t ur n i m pli es t h at sy = cy f or all n.

4 2 5 y=Y(I) and then set the inco me tax T(I) equal to I – w(Y(I)), where again w (y ) is d e fi n e d by ( 5 ). I n divi d u als will s el e ct i n c o m e s o as t o m a xi miz e I – T (I ) 2 0 – c(Y(I), n) = w(Y(I) – c(Y(I), n), by setti ng w = cy w hi c h is t h e d esir e d r es ult. It is i nteresti ng f or t h ose of us w h o live i n t he U nite d States t hat t o a first crude approxi matio n, our educatio n costs are roug hly si milar i n for m to t he efficiency-inducing tax schedule. Lo wer levels of education are subsidized a n d t hese s ubsi dies decli ne at college a n d u niversity levels beca use of t he ex- iste nce of a large private sect or i n hig her e d ucati o n.

SEC OND-BEST OPTI MA WIT H ASY M METRIC INF OR MATI ON We t ur n no w to a brief exa mi natio n of t he effects of i nvest me nt i n e d ucatio n a nd o n net i nco mes of maxi mizi ng so me social welfare fu nctio n. As before, N = w (y ) – c (y, n ) is n et i n c o m e a n d n is distri b ut e d i n t h e p o p ul ati o n a c c or d- i n g t o f ( n ). L et V ( N ) b e t h e s o ci al v al u e of n et i n c o m e of N t o a ny i n divi d u al. For t he mo me nt, we will place no restrictio ns o n V( N) ot her t ha n t hat it is u p- w ar d-sl o pi n g – m or e n et i n c o m e is b ett er. W e will a d o pt a n a d ditiv e s o ci al w el- fare f u ncti o n

.

For the mo ment, we will make no assu mptions about the shape of V( N) ex- ce pt t hat it is u p war d-slo pi ng. T he objective is to maxi mize Z s ubject to t wo co nstrai nts. O ne is t hat i n divi d uals make a ratio nal c hoice of e d ucatio n

w ( y) = cy( y, n ). T he seco nd co nstrai nt is t hat gross wages add up to total output or produc- tivity.

( 6 ) .

Let us suppose t hat t he particular fu nctio n w(y) t hat is c hose n i nduces a c hoice of y = r( n) for eac h level of t he u nobserve d c haracteristic. Differe n- tiati ng N(y, n) t otally wit h res pect t o n gives

( 7 ) .

L et N ( n ) = K . I ntegrati ng (7) wit h res pect to n we have

( 8 ) .

Noti ng t hat w = N + c, a n d s ubstit uti ng i n (6), t he co n ditio n t hat total wages eq ual total pro d uctivity across t he w hole po p ulatio n, we have

2 0 T his a nalysis ass u mes t hat t here are no a dverse i nce ntive effects of a n i nco me tax i n ter ms of t he c hoice bet wee n work a n d leis ure. If t here are s uc h effects, t he n taxi ng i nco me a n d taxi ng t he sig nal are not eq uivale nt i n ter ms of t he o utco me.

4 2 6 ( 9 ) .

T h e p oi nt of all t his is si m ply t o g et ri d of t h e f u n cti o n w (y ). T h us w e i m a gi n e selecti ng a n up ward-slopi ng fu nctio n y = r( n) to maxi mize Z. By usi ng (8) a nd (9) we are ass ure d t hat bot h co nstrai nts i n t he proble m are satis fie d, a n d t he wage or tax sc he d ule is no w here i n sig ht. We ca n calc ulate it later usi ng t he o pti mal sc he d ule r( n) by i nverti ng t hat sc he d ule to n = h(y), s ubstit uti ng for n i n w = cy a n d i ntegrati ng to get t he w(y) t hat i n d uce t he sig nal c hoices t hat s olv e t h e o pti mizi n g pr o bl e m. T h e s ol uti o n t o t h e pr o bl e m is 2 1

( 1 0) .

If V ( N ) is a c o nst a nt s o t h at V ( N ) is li n e ar, t h e n t h e ri g ht si d e of ( 1 0 ) is z er o. T his is t he case we have j ust looke d at, w here total net i nco me is maxi mize d by i n d uci ng t he ef ficie nt c h oices of i nvest me nt i n e d ucati o n. It is j ust a s pecial case of t he more ge neral proble m a nd i n t hat case, t he opti mal sc hedule is de fi ne d by sy = cy, f or all n. If V( N) is very c o ncave s o t hat t he first derivative falls ra pi dly, t he n at t he li mit t he o nly t hi ng t hat matters is t he net i nco me of t he peo ple wit h t he lo w- est net i nco me. T his case is nor mally called t he maxi mi n case. I n t his case (10) beco mes

(11) ( sy – cy) f = – cy n ( 1 – F ). T he rig ht si de of (11) is positive, w hic h mea ns t hat i nvest me nt i n e d ucatio n is belo w t he ef ficie nt a n d f ull i nfor matio n level. At t he to p of t he ra nge of n, (1–F)/f approac hes zero a nd t he i nvest me nt i n educatio n is ef ficie nt. 2 2 T he maxi mi n o utco me i n ter ms of i nvest me nt i n t he sig nal is also t he res ult t hat would obtai n if t here were a mo nopso nist purc haser of labor services. The reason is that the monopsonist wants to maxi mize the difference be- t wee n pro d uctivity a n d gross wages, s ubject to t he co nstrai nt t hat t he net i n- c o m e of t h e i n divi d u als wit h t h e l o w est n et i n c o m e n ot f all b el o w s o m e pr e- speci fied level. T hus t he solutio n to t he mo nopso nist’s proble m is to maxi- miz e t h e n et i n c o m e of t h e l o w est l ev el a n d t h e n t a k e it all a w ay by l o w eri n g w(y) u nifor mly across all e d ucatio n levels. Mat he matically yo u are j ust rever- si ng t he objective f u nctio n a n d o ne of t he co nstrai nts.

2 1 T he derivatio n of t his is a a not her a p plicatio n of t he calc ul us of variatio ns. Yo u dis place t he f u ncti o n r( n) by δ θ ( n ), differe ntiate t he w h ole t hi ng wit h res pect t o δ , a n d set t he res ult e q ual t o zero. After so me reversi ng of orders of i ntegratio n, you e nd up wit h a n equatio n of t he for m Si nce t his m ust hol d for all f u nctio ns α ( n ), t he o pti mizi ng co n ditio n is Q (r ( n ), n ) = 0. T h e a p pli c ati o n of all t h at t o t his c as e yi el ds c o n diti o n ( 1 0 ). 2 2 T his f oll o ws fr o m t he fact t hat (1-F) /f is t he reci pr ocal of t he derivative of -l og(1-F).

4 2 7 T he opposite extre me occurs w he n t he welfare fu nctio n is extre mely co n- v e x s o t h at its sl o p e ris es r a pi dly. I n t h e li mit t h at w o ul d m e a n v al ui n g o nly t h e net i nco mes of t hose wit h t he hig hest net i nco mes, t he maxi max case. I n t his case, t he opti mizi ng co nditio n (10) beco mes

(12) ( sy – cy) f = cy n F . I n t his case, t he rig ht si de of (12) is negative, i m plyi ng t hat like t he market equilibriu m wit h i mperfect i nfor matio n, t he i nvest me nt i n educatio n goes be- yo n d t he poi nt at w hic h t he directio n co ntrib utio n to pro d uctivity is reac he d. At t he botto m e n d of t he ra nge of n, F /f a p proac hes 0 a n d t h us at t he lo west levels t he i nvest me nt i n e d ucatio n is ef ficie nt. 2 3 For the cases where v( N) is not approaching an extre me (very convex or co ncave), o ne ca n see fro m (10) t hat t he rig ht side approac hes zero w he n n approaches its mini mu m and its maxi mu m values. Provided that f(n) does n ot g o t o zer o at t he extre mes, t his w o ul d mea n t hat at t he e n d val ues of n, i n- vest me nt i n e d ucati o n is ef ficie nt, t hat is sy – cy = 0. T his is g e n er ally n ot a f e a- ture of t he co mpetitive equilibriu m, a nd t hus we may co nclude t hat t he co m- p etitiv e e q uili bri u m is g e n er ally n ot t h e s ol uti o n t o s o m e o pti mizi n g pr o bl e m wit h t his f or m of s o ci al w elf ar e f u n cti o n. It is p ossi bl e t o r e writ e t h e o pti miz- i n g c o n diti o n ( 1 0 ) i n t h e f oll o wi n g f or m.

w here E(*|–) is t he co nditio nal expected value. For co ncave fu nctio ns t he ter m i n t he large sq uare brackets o n t he rig ht is al ways positive a n d t h us i n- vest me nt i n e d ucatio n is belo w t he ef ficie nt level for all n exce pt possibly at t he e n d poi nts. For co nvex welfare f u nctio ns, t he derivative is risi ng a n d t h us t he ter m i n sq uare brackets is negative. T he levels of i nvest me nt i n e d ucatio n excee d t he ef ficie nt o nes. 2 4 If y o u st a n d b a c k fr o m all t his, t h e g e n er al p att er n is f airly cl e ar. T h e m ar- ket equilibriu m produces overi nvest me nt i n t he sig nal because of t he sig nal- i ng effect, w hic h is a private be ne fit t o t he i nvest or, b ut yiel ds n o s ocial be ne- fit as its f u nctio n is p urely re distrib utive. T he re distrib utio n occ urs i n t he

2 3 T his f oll o ws fr o m t he fact t hat F /f is t he reci pr ocal of t he derivative of l og(F). 2 4 T h os e f a mili ar wit h o pti m al t a x pr o bl e ms will r e c o g niz e t h e f or m of t his o pti mizi n g c o n diti o n. - I n t h e i n c o m e t a x pr o bl e m, i n divi d u al w elf ar e m e as ur e d i n d oll ars is u ( w,l ) = wl – t( wl ) – h ( l – l), - w h er e w is i n c o m e p er h o ur w or k e d, l is w or ki n g h o urs a n d t ( wl ) is a t a x o n i n c o m e, a n d l is a c o nst a nt- o n e c a n t hi n k of it as t h e t ot al ti m e av ail a bl e. I n divi d u als m a xi miz e u wit h r es p e ct t o l by s etti n g w – t w – h = 0. T h e g ov er n m e nt h as a r ev e n u e c o nstr ai nt s o t h at If v ( u ) is t h e w elf ar e f u n cti o n t h e n w e w a nt t o m a xi miz e H er e y o u c h a n g e t h e o pti mizi n g s c h e d ul e t o y ( w ), g et ri d of t a x f u n cti o n t (y ) by n oti n g t h at i m p os e t h e r ev e n u e c o n- strai nt to deter mi ne u(0), a nd t he displace t he fu nctio n y( w) to fi nd t he opti mu m. T he opti mu m is a c hi ev e d w h e n N ot e t h at if v ( u ) is li n e ar, t h e n t he rig ht si de of t his eq uatio n is zero. Work leis ure c hoices are ef ficie nt a n d t he gover n me nt rev- e n ue is raise d by a l u m p s u m tax, w hic h does not distort t he work-leis ure c hoice.

4 2 8 directio n of i ncreasi ng t he gross a n d net i nco mes of t hose wit h hig her levels of educatio n a nd productivity a nd he nce i nco me, t he market equilibria te nd to look more like t he solutio ns to a seco nd-best opti mizi ng proble m wit h a co nvex social welfare f u nctio n. T hese, not s ur prisi ngly, are t he welfare f u nc- tio ns t hat weig ht hig her net i nco mes more hig hly t ha n lo wer net i nco mes.

T HE CASE OF ED UCATI O N C OSTS RISI NG WIT H T HE U NSEE N ATTRIB UTE: SIG NALI NG C OSTS VARY T HE WR O NG WAY WIT H RESPECT T O PR OD UCTIVITY T he sta n dar d case of sig nali ng i n w hic h t he sig nal has t he ca pacity to s urvive a n d retai n its i nfor matio nal co nte nt occ urs w he n t here is a n u nobservable at- tribute t hat is valuable to buyers (i n t he exa mples we are looki ng at, e m- ployers), a n d t he costs of u n dertaki ng so me activity t hat is observable are ne- gatively correlated with the valued attribute. The labor market exa mples, ho wever, are slig htly more co mplicated i n t hat i n t hat t he u nobserved at- tribute co ntributes to t he i ndividual’s productivity a nd so does t he sig nal. T h us, w hatever t he u nobservable attrib ute is, it has t wo so urces of val ue. O ne is t he direct effect o n pr o d uctivity a n d t he ot her is t hat l o weri ng t he c osts of acq uiri ng h u ma n ca pital also has val ue. U p to t his poi nt we have ass u me d t hat all of t his w orks i n t he sa me directi o n. B ut it is p ossi ble t hat attri b utes t hat lo wer t he cost of acquiri ng educatio n mig ht not be t hose t hat e n ha nce pro- d uctivity or mig ht eve n be attrib utes t hat have a negative effect o n pro d ucti- vity. W e k n o w t h at if S n = 0, t h e n y o u c a n g et a si g n ali n g e q uili bri u m, t h o u g h i n t hat case t he sig nal is n’t needed: it si mply ide nti fies ex post t hose wit h lo wer educatio n costs a nd he nce hig her levels of educatio n.

T h e q u esti o n I p os e n o w is, c a n y o u g et a si g n ali n g e q uili bri u m w h e n cy n > 0, ass u mi ng t hat sn > 0? Fro m t he a nalysis of eq uilibria, we k no w t hat t he seco n d order co nditio n for t he i ndividual’s c hoice of educatio n is satis fied o nly if

T his m e a ns t h at if cy n > 0 o n e c a n h av e a n e q uili bri u m wit h si g n ali n g o nly if S o t h e q u esti o n is, c a n t h at h a p p e n? T h e o nly w ay it c o ul d h a p p e n is if t h e h u m a n c a pit al eff e ct is l ar g e i n t h e s e ns e t h at it overri des t he negative sig nali ng effect. T h us if sy 0, it c ert ai nly c a n n ot h a p- pe n. T he sig nal makes no direct co ntributio n to productivity a nd se nds t he wro ng message about t he u nobserved attribute. Everyo ne will set y = 0. But t he a ns wer to t he questio n posed above is yes, provided t hat t he hu ma n capi- tal effect is large e no ug h to overri de t he negative sig nali ng effect. T here ca n be sig nali ng eq uilibria i n w hic h sig nali ng costs rise wit h t he level of t he u n- observe d attrib ute t hat co ntrib utes to pro d uctivity. I will de mo nstrate t his by exa mple i n a mo me nt. I ntuitively t his s hould make se nse if t he hu ma n capital effect is big e n o ug h, beca use t he n t he attri b ute t hat is of real val ue t o t he i n- divi d ual a n d to t he e m ployer is t he o ne t hat drives e d ucatio n costs do w n. T he key is t hat t he wage sc he d ule has to be u p war d-slo pi ng i n t he sig nal, a n d t his c a n h a p p e n if sy is bi g e n o u g h ev e n if sn > 0. I n t his c o ntext, t he c orrect state- me nt abo ut t he co n ditio n t hat allo ws for a sig nali ng eq uilibri u m is t hat t he

4 2 9 net be ne fit of acquiri ng t he sig nal has to be positively correlated wit h t he gross effect o n pro d uctivity. T his co ul d ha p pe n i n seg me nts of t he po p ulatio n if very tale nted people face hig h opportu nity costs associated wit h spe ndi ng ti m e o n e d u c ati o n. T his case may or may not be i nteresti ng fro m a n e mpirical poi nt of vie w, b ut it does ill ustrate t hat t he more ge neral for m ulatio n of t he co n ditio ns for sig nali ng are i n ter ms of gross a n d net be ne fits a n d not j ust sig nali ng costs. I di d not realize t his w he n I first worke d o n sig nali ng eq uilibria. I t ho ug ht t he n t hat t he abse nce of t he i nt uitively pla usible negative cost correlatio n co n di- ti o n w o ul d d estr oy a si g n ali n g e q uili bri u m. It is als o of p ot e nti al i nt er est b e- ca use t he sig nali ng effect is reverse d. If cy n > 0, t h e n i n e q uili bri u m if t h er e is si g n ali n g, d n / dy < 0, a n d t h us

.

T herefore t he negative sig nali ng effect ca uses t he private ret ur n to e d ucatio n to fall s hort of t he social retur n a nd he nce causes u nder-i nvest me nt i n edu- c ati o n. 2 5 It re mai ns to s ho w by a co nstr uctive exa m ple t hat t his ca n ha p pe n at least i n t he t heory, a n d per ha ps also i n t he worl d. W e will s h o w t h at t his c a n h a p p e n wit h a n e x a m pl e. L et s( n, y ) = ny θ , a n d l et c( n, y ) = n α yβ , w here α < 1. Follo wi ng t he sta ndard equilibriu m a nalysis, we hav e

α β – 1 ( 1 3) w = cy = β n y . I n a d diti o n w = ny θ , or n = wy θ . S u bstit uti ng i n (13), we have t he differe ntial equatio n i n w(y) t hat de fi nes t he equilibriu m wage sc hedules w – α w = β y( β – 1 – α θ ).

O ne of t he s ol uti o ns t o t his differe ntial e q uati o n is 2 6

w here K is a co nsta nt. If you t he n proceed to deter mi ne equilibriu m c hoices of e d ucati o n, t hey are give n by

w here T is a not her co nsta nt. T he seco n d or der co n ditio n is satis fie d a n d t his all w or ks if dy / d n < 0 a n d t his will i n f a ct b e t h e c as e pr ovi d e d t h at α < 1, a n d β < θ . T h at is t o s ay, si g n ali n g o c c urs a n d t h er e is a s e p ar ati n g e q uili bri u m, if t he elasticity of pr o d uctivity wit h res pect t o e d ucati o n is larger t ha n t he elasti- city of e d ucatio n costs wit h res pect to e d ucatio n. Or i n si m ple or di nary la n-

2 5 I thought Professor Gary Becker whose pioneering work in hu man capital, household produc- tio n fu nctio ns a nd muc h more, mig ht appreciate t his result. 2 6 As t his is si m ply a n ill ustr ativ e e x a m pl e, t h er e is n ot m u c h p oi nt i n st u dyi n g all t h e e q uili bri a i n t h e e x a m pl e.

4 3 0 guage, educatio n is productive e noug h to justify its costs a nd override t he negative sig nali ng effect. A not her way to t hi nk about t hese relatio ns hips a nd t his case is to c ha nge t he u nobserved variable fro m n to t = n– – n , w h er e n– is t h e hi g h est l ev el of n. I n effect t his de fi nes t he u nobserved c haracteristic to be t hat w hic h lo wers e d ucatio n costs. No w i nvest me nt i n e d ucatio n will rise wit h t, provi de d t here is a se parati ng eq uilibri u m, beca use c yt < 0. B ut t he effect of t o n pr o d uctivity is no w negative, a n d if t hat effect is stro ng e no ug h relative to t he h u ma n ca- pital effect, t he n t hat will preve nt a se parati ng versi o n of t he sig nali ng e q ui- libriu m. T he reaso n is t hat u nless t he hu ma n capital effect is large e noug h, t he sig nali ng effect will ca use t he wage f u nctio n to be do w n war d-slo pi ng i n y. T h er e will b e a p o oli n g e q uili bri u m at y = 0 a n d t h at of c o urs e e nt ails i n t h e hypot hetical exa mple, reaso nably dra matic u nder-i nvest me nt i n t he pote ntial sig nal eve n t ho ug h it is pro d uctive h u ma n ca pital.

T HE INF OR MATI ON C ONTAINED IN T HE SIGNAL CAN I MPR OVE PR OD UCTIVITY It s hould be noted t hat t he i nfor matio n carried by t he sig nal ca n be produc- tive itself. T his will occ ur if t here is a decisio n t hat is ma de better or wit h greater ef ficie ncy, wit h better i nfor matio n. I n t he job market co ntext, o ne c o ul d b uil d t his i n as f oll o ws. L et pr o d u ctivity b e v ( n,y, d ) w h er e d is a d e cisi o n t hat t he e m ployer makes. It co ul d be a decisio n abo ut w hat ty pe of job to as- sig n to so meo ne or a decisio n about required trai ni ng. O ne ca n i nterpret t he prece di ng a nalysis t he n, at least f or e q uili bria i n w hic h t he sig nal carries i n- for matio n i n equilibriu m, as s( n,y) bei ng t he maxi mu m of v( n,y,d) wit h re- s pect to d for eac h level of n a n d y. Note t hat t his co m po ne nt of val ue is dif- fere nt fro m t he hu ma n capital effect. U nlike t he hu ma n capital effect, t his ele me nt of val ue o nly exists if t he sig nal is carryi ng i nfor matio n i n eq uili- bri u m. Y o u w o ul d l os e it i n a p arti al or c o m pl et e p o oli n g e q uili bri u m ev e n if t hat equilibriu m i ncluded i nvest me nt i n educatio n. In situations like the Lazear model that we exa mined earlier, in which t here is bot h pooli ng a nd separati ng co mpo ne nts of t he equilibriu m, t he ad- diti o n of a d e cisi o n t h at is m a d e b ett er wit h i nf or m ati o n will c a us e t h e v al u e of t he sig nal to rise, or more acc urately t he net be ne fits to rise. T hose net be ne fits are passe d o n to i n divi d uals by virt ue of co m petitio n o n t he e m ploy- er side. T hus t he relative sizes of t he separati ng a nd pooli ng co mpo ne nts of t h e e q uili bri u m will s hift i n f av or of t h e s e p ar ati n g p art. M or e p e o pl e will o pt to go to t he fir ms t hat i ncur t he expe nse of mo nitori ng a nd lear ni ng produc- tivity ov er ti m e.

4 3 1 TI ME A N D T HE ALL OCATI O N OF TI ME AS A SI G NAL A N D A SCREE NI NG DEVICE 2 7 T he pri nci ples t hat gover n t he s urvival of sig nals i n markets ca n be a p plie d to ot her co ntexts. T he ki n d of mixe d or i m perfectly alig ne d i nce ntive str uct ure t hat c haracterizes markets is co m mo n i n ma ny sit uatio ns. I n partic ular, sit ua- tio ns i n w hic h i ndividuals have a n u nk no w n or i mperfectly perceived level of interest in so meone or so mething are ubiquitous. In these situations, one fre q ue ntly o bserves t hat ti me s pe nt by t he i n divi d ual is take n as a sig nal of i n- terest. Literat ure a n d every day ex perie nce s uggest t hat t he allocatio n of ti me is a ubiq uito us a n d persiste nt sig nal so meti mes use d deliberately as a scree n- i n g d evi c e. T he persiste nce of s pe n di ng ti me as a sig nal of i nterest i n s o met hi ng is a re- flecti o n of t he fact t hat ti me is i n s h ort s u p ply a n d every o ne k n o ws it. T here is a no n- zero s hado w price o n ti me, a nd he nce t he use of it i n pursuit of so me perso n, goal or i nterest must mea n t hat so me i mplicit net be ne fit test has bee n passe d for t hose w ho s pe nt t he ti me a n d faile d for ot hers w ho di d not. T he use a n d i nter pretatio n of ti me as a sig nal is co m plicate d a n d e nric he d by t he acc urate observatio n t hat t here are bot h perceive d a n d act ual differ- e nces across people i n t he s hado w price o n ti me. T hese differe nces are used to i nter pret t he sig nal. T he allocatio n of a s mall a mo u nt of ti me by a n i n di- vidual wit h a hig h perceived s hado w price o n ti me has t he sa me weig ht as a larger allocatio n of ti me by a n i n divi d ual wit h a lo wer perceive d s ha do w price. As a n exa mple, t hose w ho have had visible leaders hip positio ns i n virtually a ny orga nizatio n k no w t hat t he eve nts t hey atte n d are take n as sig nals of i n- terest a nd support, a nd co nversely, w he n t hey do not s ho w up, t hey delibe- rately a nd so meti mes i nadverte ntly sig nal a lack of i nterest, support or e n- thusias m. It is note worthy that the infor mational content of these co m mon sig nals is n ot necessarily relate d t o w het her t he i n divi d ual (t he sig naler) is ac- t ually nee de d or has a r ole at t he eve nt. I n fact, havi ng a r ole at t he eve nt ca n dil ute t he sig nal beca use it co m plicates t he i nter pretatio n of t he reaso n for t he lea der’s prese nce. Ti me is also ro uti nely use d as a scree ni ng device or as part of t he price of a d missio n to eve nts at w hic h t he n u mber of places is i n s hort s u p ply. For e n- tertai n me nt a nd sporti ng eve nts it is co m mo n to fi nd t hat a mixed syste m t hat i nvolves bot h price a n d ti me is use d to allocate t he li mite d n u mber of places at t he eve nt. T he questio n t hat I address brie fly i n t his sectio n is w hy t he mixed syste m would be used i n prefere nce to t he pure price syste m. T he i n- t uitiv e a ns w er is t h at willi n g n ess or a bility t o p ay m ay b e a p o or si g n al of r e al i nterest i n t he eve nt, n ot wit hsta n di ng t he fact t hat fail ure t o use t he price sys- te m is i nef ficie nt for t wo reaso ns. 2 8 T he mixe d syste m allocates places or ac-

2 7 Part of this section is based on a nearly unkno wn (probably for good reason) paper by the author, “ Ti me and Co m munication in Econo mic and Social Interaction,” Q uarterly Jo ur nal of Eco no mics. 2 8 T here are perfor mers w ho wa nt people wit h a ge nui ne i nterest, a nd not just t hose wit h a n i n- terest a n d hig h i nc o mes at t he eve nts, a n d w h o feel t hat t he q uality of t he eve nt is affecte d by i n- teraction bet ween the perfor mer and a mong me mbers of the audience.

4 3 2 cess to t hose w ho wo ul d (a n d ofte n do if t here is a seco n dary market) sell t he place to so meo ne wit h a hig her willi ng ness to pay – w he n t hat happe ns bot h parties are better off. Sec o n d, t he ti me t hat is use d t o ac q uire t he places is a dead- weig ht loss u nder most circu msta nces. Not wit hsta ndi ng t he clear prob- le ms wit h t he fail ure to use t he price syste m, t he use of t he mixe d syste m is very co m mon, and must mean that so me objective is being pursued that is n ot res pectf ul of t he c urre nt distri b uti o n of i nc o me. F or if t he distri b uti o n of i n c o m e is vi e w e d as o pti m al, it is h ar d t o c o nj ur e u p a r ati o n al e f or i n c urri n g t he costs of t he mixed syste m me ntio ned above. I a m goi ng to use a si mple exa mple to look at t he effects of usi ng a mixed syste m for t his class of reso urce allocatio n proble ms. It is not mea nt to be a general model, but rather to suggest what para meters are i mportant in de- ter mi ni ng t he perfor ma nce c haracteristics of t he outco me. Let t he val ue attac he d to a n eve nt by a n i n divi d ual be θ . O n e c a n t hi n k of t his as a d oll ar v al u ati o n b as e d o n a r o u g hly e q u al distri b uti o n of i n c o m e, or y o u c a n t hi n k of it as a m e as ur e of v al u e i n a p ur e ti m e syst e m. W h at it is n ot is a c urre nt willi ng ness t o pay f or access t o t he eve nt. It is u nif or mly distri- buted i n t he populatio n, o n t he i nterval [0,1]. T he value of mo ney is λ . It als o is distri b ut e d u nif or mly i n t h e p o p ul ati o n o n t h e i nt er v al [ 0, 1 ]. T h e as- s u m pti o n a b o ut t h e v al u e of ti m e at t h e m ar gi n is t h at it is e q u al t o G – α λ . If t h e p ar a m et er a is z er o, t h e n it is a c o nst a nt, if a > 0, t h e n t h e m ar gi n al v al u e of ti me is negatively correlated wit h t he margi nal value of i nco me, a nd t he co nverse is also tr ue. T he fractio n of t he po p ulatio n t hat ca n have access to t h e ev e n is N, a n d w e will ass u m e t h at N < 0. 5. Fi n ally l et p b e t h e m o n et ar y price f or access t o t he eve nt a n d let T be t he ti me price. The set of people who attend the event is the people with co mbinations of ( θ , λ ) t hat satisfy t he i ne q uality θ – λ p – ( G – α λ ) T = θ – ( p – α T ) λ – G T ≥ 0. L et S = G T a n d R = p – a T . T he n t he i nequality above beco mes θ – R λ – S ≥ 0 . L et A = { ( θ ,λ ) ): θ – R λ – S ≥ 0}. T he n beca use of t he fixe d s u p ply, t he area of t he regio n associate d wit h A has to be N. Fi g ur e 7 is us ef ul i n a n alyzi n g t h e c h oi c es. E a c h of t h e li n es i nsi d e t h e u nit square represe nt co mbi natio ns of R a nd S, so t hat t he area above t he li ne is N. T he li ne t hro ug h t he origi n re prese nts t he p ure price syste m. As t he ti me price T is raise d, S rises a n d R falls. It is easy to see wit ho ut a co m plicate d pr o of t hat as T rises, t he average val ue of θ is risi n g w hil e t h e t ot al n u m b ers stay t he sa me, s o t hat t he t otal gr oss val ue t o c o ns u mers is risi ng. T his c o nti- n u es u ntil t h e li n e g o es fl at, at w hi c h p oi nt R = 0. It is als o cl e ar t h at t h e i n- cr e as es ar e l ar g est f or l o w l ev els of T or S, b e c a us e i niti ally o n e is tr a di n g l o w l ev els of θ f or m u c h hi g h er l ev els. It is possible t hat t he mixe d ti me a n d price syste m is co m mo nly use d si m ply to ac hieve t his result, na mely allocati ng t he spots to t hose w ho attac h t he hig hest value to t he m, a nd t hat w hoever makes t hese decisio ns does n’t t hi nk

4 3 3 Fi g ur 7 about or care about t he dead weig ht cost of t he ti me co mpo ne nt of t he price. B ut perha ps a more i nteresti ng q uestio n is w het her t he use of t he mixe d sys- te m ever i ncreases t he net be ne fits. We will ass u me t hat t he dollar reve n ue is distrib ute d i n a ne utral way so t hat price co m po ne nt of t he access c harge ge- nerates neither net benefits or costs. The ti me co mponent of the access c harge ho wever ge nerates a dead weig ht cost. T he net be ne fits are

T he seco nd t wo ter ms are t he dead- weig ht cost of t he ti me co mpo ne nt of t he access c harge. We have alrea dy observe d t hat t he first ter m, t he gross be ne fits, ar e a risi n g f u n cti o n of S = G T, r e fl e cti n g t h e us e of ti m e as p art of t h e all o- catio n mec ha nis m. T his is true i ndepe nde nt of t he sig n or size of t he para- meter “a”. Moreover, o ne ca n see fro m Figure 7 t hat starti ng at S = 0, t he gai ns fr o m i ncreasi ng S, or T, are largest at t he start (t hat is near S = 0) be- ca use as t he li nes rise a n d r otate rig ht, l o w average levels of t he val uati o n pa- ra meter θ are being knocked out and high ones are being brought in. The differe nce bet wee n t he exits a n d t he e ntries stea dily decli nes as S i ncreases, a n d t he li nes beco me flatter. T his bri ngs us to t he q uestio n of w het her t he net be ne fits also i ncrease at least over so me ra nge as s i ncreases. It is so me w hat te dio us b ut not dif fic ult to s h o w t hat i n t his exa m ple t hat t he sl o pe of gr oss be ne fits at S = 0 is

4 3 4 .

T he derivative of net be ne fits at S = 0 is 2 9

If t he para meter a is zero, t he n o ne ca n see t hat i n t his exa mple, t he dead- weig ht costs o ut weig h t he gai n i n gross be ne fits. Ho wever if “a” is positive a n d large e noug h, t he n t he net be ne fits rise at least over so me ra nge. W hy s hould t his o c c ur i nt uitiv ely? H er e pri c e is a b a dly fl a w e d si g n al of v al u e. Ti m e, if its margi nal cost is u ncorrelate d wit h price, is a better sig nal of val ue, b ut not good enough to overco me the dead- weight cost. But if the marginal cost of ti me is negatively relate d to t he margi nal val ue of i nco me, t he n t he co mbi na- ti o n of pri c e a n d ti m e is a m u c h b ett er si g n al of v al u e. A n d if m ar gi n al ti m e costs fall quite rapidly as t he margi nal value of i nco me i ncreases, t he n t he c ost of t h e ti m e t h at is n e e d t o scree n o ut t he hig h i nco me-lo w value purc hasers is r el ativ ely l o w. To ill ustrate, Fig ure 8 s h o ws t he val ue of net be ne fits as a f u ncti o n of S, f or a c as e i n w hi c h G = 2, N = . 2 5 a n d a = 1 4. H er e t h e r e a d er s h o ul d i g n or e t h e hig her levels of S, as t hey are ass ociate d wit h negative prices. A b ove a certai n level of S, or ti me cost, t he model starts redistributi ng i nco me, w hic h is c o u nter t o t he s pirit of t he a nalysis. 3 0 To su m marize, w hile t his si mple exa mple hardly disposes of t he issue of ho w markets go about tryi ng to solve co mplex allocatio n proble ms i n a world of i m perfect i nfor matio n, it does, I ho pe, s uggest t hat t he ki n ds of mixe d sys- te ms t hat we observe i n reality may be quite i n novative solutio ns to t hese proble ms a nd probably s hould not be dis missed i n all cases as mistakes made by people who don’t understand the virtues of the price syste m. There may eve n be more i nteresti ng a nd less wasteful alter native curre ncies, suc h as ho urs s pe nt i n p ublic service work, t hat co ul d be use d i n co nj u nctio n wit h t he price syste m to allocate hig hly val ue d access to i m porta nt eve nts. W hile s uc h alter natives mig ht n ot be as effective as ti me i n u n d oi ng t he effect of t he price syste m, t hey may also have co nsi derably less dea d weig ht cost associate d wit h t h e m.

2 9 T hese res ults follo w fro m t he follo wi ng facts. For R + S > 1, gross be ne fits are

a n d t he dea d weig ht ti me c ost is Si milarly, w h e n R + S < 1, w hi c h o c c urs as S b e c o m es l ar g er, R = 2 ( 1 – S – N ), gross be ne fits are (1 /2)[(1 – S 2 ) – R S – ( R 2 / 3 ], a n d t h e d e a d w ei g ht c ost of t h e us e of ti m e is W h e n R + S = 1, t h e li ne t hat separates t hose w ho are ad mitted fro m t hose w ho are not goes t hroug h t he poi nt (1,1) w hic h is w hy t he f or m ulae s hift at t hat p oi nt. 3 0 T he existe nce of seco ndary market will to so me exte nt reduce t he be ne fit of t he mixed syste m because it will dra w i nto to pri mary allocatio n syste m e ntrepre neurs w ho are t here to use t heir r el ativ ely l o w v al u ati o n o n ti m e t o m a k e a n ar bitr a g e pr o fit. T h ey will bi d u p t h e pri c e r e q uir e d to clear the pri mary market and hence drive so me of those with high gross valuations of the eve nt fro m t hat market.

4 3 5 Fi g ure 8

T HE INTERNET AND T HE C HANGING INF OR MATI ONAL STRUCTURE OF MARKETS T he pr o p ositi o n t hat t he i nter net, or m ore acc urately its relatively rece nt ac- cessibility to a broa d s pectr u m of users, has c ha nge d t he i nfor matio nal str uc- ture of ma ny markets, i ndustries a nd eco no mies, is probably not co ntrover- sial. One could argue about how rapid the change has been fro m quite sudden on the one hand to a gradual evolution fro m the telegraph ( which was t he first ti me t hat co m mu nicatio n over dista nce did not require t he p hy- sical move me nt of so met hi ng a nd he nce beca me i n a certai n se nse nearly i n- sta nta neo us) t hro ug h ra dio, t he tele p ho ne, televisio n, fax etc. o n t he ot her. It h as als o b e e n e asy f or s o m e t o dis miss all of t his as a f a d, r elyi n g o n t h e b o o m- a nd-bust cycle t hat we have just observed. But i n my judg me nt t hat would be a mistake. We k no w fro m t he eco no mics of i nfor matio n t hat t here are periods i n w hic h e noug h has c ha nged so t hat we operate i n a data-free e nviro n me nt, t hat is, o ne i n w hic h t here is te m p orarily little or n o releva nt data t o c o nstrai n ex pectatio n. T he data trickle i n a n d t he beliefs a n d ex pectatio ns start to li ne u p wit h r e ality a g ai n. T he more fu nda me ntal poi nt is t hat i nvestors a nd ot hers were probably not wro ng about t he ulti mate effect of t his tec h nology o n markets a nd t he eco- no my, but al most everyone overesti mated the speed with which individuals and organizations change their behavior, and we also underesti mated the a m o u nt of dif fi c ult t e c h ni c al i nfr astr u ct ur e t h at n e e d e d t o b e b uilt i n or d er to have the foreseen outco mes beco me a reality. We would undoubtedly have bee n better off if we had re mi nded ourselves of t he i mporta nt work of t he late Zvi Grilic hes o n t he diffusio n of i n novatio n, as t hat would have caused so me questioning of the assu mption that what very intelligent people can

4 3 6 foresee will co me about quickly. I wa nted also to recog nize Zvi’s work as a n early a nd very i mporta nt exa mple of be havioral eco no mics. Nevert heless t here are po werful forces drivi ng t he outco mes a nd c ha ngi ng the infor mational structure of markets. Three of the most i mportant are Moore’s la w, Metcalfe’s la w a n d t he dra matic re d uctio n of t he noise to sig nal ratio n i n fiber o ptic cable res ulti ng i n very large ex pa nsio ns i n t he t hro ug h- put capacity of t hese pipes measured i n sig nal processi ng ter ms. Moore’s la w is well-k no w n. It is a n e m pirical observatio n t hat t he n u mber of tra nsistors o n a chip doubles every eighteen months to t wenty-four months. To a first ap- proxi matio n t his has produced roug hly a 10-billio n-ti mes cost reductio n i n t he first fifty years of t he co mputer age, dated fro m about 1950. Or to put it a not her way, t hi ngs t hat were i magi nable but u ni magi nably expe nsive i n 1950 are esse ntially c ostless t o day. Ec o n o mic hist oria ns will be better a ble t ha n I t o say w het her t here were co mparable periods of c ha nge i n t he past wit h respect to costs of doi ng so met hi ng i mporta nt. Metcalfe’s la w states t hat t he val ue of a net w ork t o t he e ntities attac he d t o it is proportio nal to t he square of t he nu mber of co n nected e ntities. 3 1 I n ec o- no mic ter ms t his probably mea ns t hat t he value a nd he nce t he speed of co n- necti ng accelerates as t he n u mbers i ncrease. T his is so meti mes referre d to as t he net w ork effect. Fro m sig nal processi ng t heory we k no w t hat t he capacity of a c ha n nel is proportio nal to t he log of o ne plus t he ratio of sig nal to noise. T hroug h scie nti fic a n d tec h nical a dva nces (i n a d ditio n to bei ng able to use lasers wit h m ulti ple wavele ngt hs), t he noise ge nerate d w he n t he lig ht re flects off t he si de of t he fibero ptic cable is bei ng re d uce d dra matically, ca usi ng very large i n- creases i n data rates o n existi ng fiber. I n eco no mic ter ms t his tra nslates i nto large cost reductio ns i n providi ng ba nd widt h. T hese t hree effects i nteract wit h eac h ot her over ti me to produce accele- rati ng eco no mic effects. Muc h of t he rece nt eco no mic i mpact of t he i nter net is ass ociate d wit h re d uci ng tra nsacti o n c osts of a variety of ki n ds. I will ret ur n to t his s ubject i n a mo me nt. For most of t hese effects, o ne nee ds not j ust co m- p uti ng po wer, b ut also a reaso nably ubiq uito us reliable net work t hat has sta n- dar dize d protocols, is more or less al ways o n. After all, tra nsactio ns i nvolve, al most by definition, more than one party, though increasingly one or more of t he parties are mac hi nes. It is i nteresti ng a n d n ot s ur prisi ng t heref ore, t hat for t he first forty years of t he proliferatio n of co mputers, not muc h measur- able productivity i ncrease was detectible i n macroeco no mic data. I n t he past te n years, t he period i n w hic h t he expa nsio n of t he net work occurred, t here are n oticea ble meas ura ble pr o d uctivity gai ns. It see ms very likely t hat t he pr o-

3 1 T his s hould be t houg ht of as a n e mpirical regularity as well. It ca n be derived as follo ws. S u p pose t hat t he val ue a perso n to eac h ot her perso n o n a net work is x. If t here are n peo ple o n the net work and we add another person, the n people experience added value of nx and the ne w pers o n ex perie nces val ue of nx. If V( n) is t he val ue of a net w ork, bei ng t he s u m of its val ue t o all co n nected people, t he n V( n + 1) = V( n) + 2 nx. T herefore V( n) = 2x[1 + 2 + ... + ( n-1)]. Usi ng Ga uss’s for m ula for t he s u m of t he i ntegers fro m o ne to n, we have V( n) = 2x n( n-1), w hic h is t he q ua dratic relati o ns hi p t hat Metcalfe’s la w asserts.

4 3 7 d uctivity gai ns (t he o nes we ca n detect) are associate d wit h re d uce d tra nsac- tio n costs, i mproved perfor ma nce of markets, t he ability to create ne w mar- kets t hat were too expe nsive to create wit hout t he tec h nology, a nd t he i m- porta nt ability to take ti me a nd cost out of t he coordi natio n of eco no mic a ctivity, i nsi d e t h e fir m a n d i n t h e s u p ply or v al u e- a d d e d c h ai n. T h at is w h at t he net work, wit h hig h a nd reliable capacity, wit h gro wi ng nu mbers co n nec- ted to it and with enough co mputing po wer to run the endpoints and the no des has per mitte d. It is a c u m ulative effect t hat we are j ust begi n ni ng to s e e. B uil di n g m o d els i n e c o n o mi cs is a n art a n d a s ci e n c e. T h e s ci e n c e p art c o n- sists of deter mi ni ng a nalytically t he co nseq ue nces of t he ass u m ptio n t hat cre- ate t he str uct ure of t he mo del. T he art co nsists i n deci di ng w hat to p ut i nto t he structure a nd w hat to leave out. Putti ng i n too muc h makes t he models i n- tractable a n d of little use i n ill u mi nati ng t he deter mi na nts of market perfor- ma nce. P utti ng i n eit her too little or t he wro ng str uct ural feat ures makes t he res ults, t ho ug h tractable, u ni nteresti ng. A nat ural co nseq ue nce of t his is t hat para meters that don’t change much fro m market to market or over ti me tend to be suppressed, as a matter of good practice i n applied microeco no mic t he- or y. I m e nti o n all of t his b e c a us e it is p ossi bl e a n d ev e n li k ely t h at s o m e p ar a- meters relate d to searc h costs, tra nsactio n costs, acq uiri ng i nfor matio n, a n d geogra p hy have s hifte d reaso nably ra pi dly i n t he past fe w years (or are i n t he process of s hifti ng), as a res ult of t he i ncreasi ng s pee d, ubiq uity a n d co n nec- tivity of t he i nter net. T his pote ntial s hift i n para meters creates t he o p port u- nity to revisit as pects of t he i nfor matio nal str uct ure of markets a n d orga niza- tions, and in a sense, reintroduce the suppressed para meters as they have beco me i nteresti ng agai n. I wo ul d like to co ncl u de t his essay by s uggesti ng a fe w areas i n w hic h s uc h a n i nq uiry mig ht yiel d so me i nteresti ng res ults. O ne mig ht have t he i mpressio n fro m a first course i n eco no mic t heory t hat markets co mbine supply and de mand curves to produce prices, quantities a nd to deter mi ne w ho buys t he good. T here is not hi ng wro ng wit h t hat vie w. B ut i n a d ditio n, a market eco no my perfor ms lots of ot her f u nctio ns. Pote ntial buyers a nd sellers need to fi nd eac h ot her. Ofte n buyers a nd sellers need to a c q uir e i nf or m ati o n a b o ut e a c h ot h er a n d a b o ut t h e pr o d u ct. If t h e b uy ers, sellers, a nd products are differe ntiated, t he n t here is a matc hi ng proble m t hat i n o ne for m or a not her nee ds to be resolve d i n t he market place. If sellers are c hargi ng differe nt prices, t he n buyers need to co nsider e ngagi ng i n so me sort of reaso nable searc h for t he lo wer prices. T hese fu nctio ns are not per- for med costlessly, and the costs of perfor ming the m are frequently lu mped toget her u n der t he hea di ng of tra nsactio n costs. Broa dly t hese costs are bei ng c ha nged a nd lo wered by t he i nter net.

B UYERS A ND SELLERS FI NDI NG EAC H OT HER Probably t he most obvio us exa m ple of re d uce d tra nsactio n costs is t hat asso- ciate d wit h t he ra pi d ex pa nsio n of t he markets for collectibles a n d use d pro- d ucts of all ki n ds. I n t his are na t he over w hel mi ngly largest market maki ng e n-

4 3 8 tity is e Bay. T here are o n average 750,000 tra nsactio ns or eac h day wit h a n av- erage total daily dollar vol u me of 30 millio n. 3 2 Most of t hese markets si m ply di d not exist previo usly, a n d t hose t hat di d exist were more costly to create, less ef ficie nt, a n d less liq ui d. It is wort h noti ng t hat t he dro p i n t he cost of b uyers a n d sellers fi n di ng eac h ot her is largely i n de pe n de nt of p hysical geo- graphy (recall that both distance and ti me are si multaneously co mpressed wit h t he i nter net tec h nology). T his partial re moval of t he geogra p hic bo u n ds of markets makes t he m more liquid a nd i n a certai n se nse more co mpetitive. Ho wever, t here are ele me nts of natural mo nopoly i n t he market- maki ng fu nc- tio n. As t he nu mber of bidders i ncreases t he expected wi n ni ng bid rises i n t hese a ucti o ns. He nce t he market place t hat is i n t he lea d will attract t he sell- ers, a n d t he variety of pr o d ucts will attract t he b uyers. 3 3 Fig ure 9 s h o ws t he re- latio ns hip bet wee n t he nu mber of bidders a nd t he expected selli ng price i n a n or di nary a uctio n.

Fi g ure 9

3 2 Figures fro m Jeff Skoll, cofou nder of e Bay. T here are of course more markets ope n i n a ny giv- e n day t ha n t here are tra nsactio ns. 3 3 T here is a pote ntial q uali ficatio n to t his te n de ncy. I ntellige nt soft ware age nts are i n pri nci ple ca pable of searc hi ng for lo w prices across market places. If i n divi d uals were pre pare d to use t hese age nts, t hat would eli mi nate t he liquidity adva ntage of t he larger market- makers. Ho wever, t his does not a p pear to have ha p pe ne d, at least not to t he exte nt t hat it sig ni fica ntly di mi nis hes e Bay’s a dva ntage. T he maki ng of o nli ne markets is pretty clearly a pote ntial area for researc h.

4 3 9 SEARC HI NG F OR T HE L O WEST PRICE T he late George Stigler recog nize d t hat fi n di ng t he lo west price was a n acti- vity t hat required resources a nd t hat t here was a tradeoff bet wee n i ncurri ng costs of f urt her searc h a n d t he ex pecte d be ne fits of fi n di ng eve n lo wer prices. For prices t hat are poste d i n a n i nter net e nviro n me nt, t he cost of fi n di ng t he lo west price is pretty close to zero. I n pri nci ple, t his s ho ul d eli mi nate price dis persio n by eli mi nati ng o ne si de of t he tra deoff. T here was a ki n d of nat u- ral partial protectio n fro m price co mpetitio n, t he mag nitude of w hic h is a fu nctio n of t he searc h costs. T he reductio n or eli mi natio n of t hese searc h costs i n t he first i nsta nce i ncreases co mpetitio n. Ho wever t here is probably m or e t o t his st or y. T h e d e cisi o n t o p ost a pri c e is a str at e gi c d e cisi o n a n d i n t h e f a c e of n e gli gi bl e s e ar c h c osts, it is p ossi bl e t h at s ell ers’ willi n g n ess t o p ost prices will decli ne a n d t hat t here will be m ore neg otiate d prices or prices t hat ar e t ail or e d t o t h e i n divi d u al b uy er.

T HE B O U NDARIES OF T HE FIR M T here is a well-k no w n a nd i mporta nt literature i n eco no mics associated wit h R o n al d C o as e a n d Oliv er Willi a ms o n a n d ot h ers. It d e als i n p art wit h t h e f u n- da me ntal questio n of t he questio n of w hat eco no mic processes are co ntai ned wit hi n a fir m a nd w hic h are mediated by markets, t hat is, tra nsactio n bet wee n fir ms. O ne as pect of t his ge neral set of q uestio ns has to do wit h o utso urci ng: t he decisi o n t o c o n d uct a certai n set of activities i n- h o use or t o c o ntract wit h a not her e ntity to have t he m co nducted for t he fir m. Ge nerally o n t he side of o utso urci ng is t he likeli hoo d t hat certai n f u nctio ns ca n be perfor me d better by specialists who have the advantages of econo mies of focus and scale. Cou nteri ng t hat are t he co ntracti ng, mo nitori ng a nd i mple me ntatio n costs of havi ng a seco n d party provi de t he service. So me of t hese co u ntervaili ng costs are j ust tra nsactio n costs associate d wit h co m plexity i n co m m u nicatio n. It is beco ming reasonably clear that the internet platfor m is reducing so me of t hese tra nsactio n costs a n d he nce ti p pi ng t he bala nce i n a n u mber of areas i n t he directio n of outsourci ng. Ma ny, if not most, i nfor matio nally based ser- vices are ef ficie ntly deliverable a n d mo nitorable o n t he platfor m. We have i n a se nse, for t he period of tra nsitio n, a natural laboratory i n w hic h tra nsactio n cost para meters are bei ng s hifte d, res ulti ng i n ne w ex peri- me ntal be havior o n t he part of fir ms. I n ge neral, t he ra nge of eco no mic acti- vity that can be effectively coordinated across a co mplex multifir m supply c hai n is just starti ng to be explored by co mpa nies a nd t hose w ho do researc h o n t hese s ubjects. T he bo u n daries of t he fir m, tra nsactio n costs, s u p ply c hai n arc hitect ure a n d coor di natio n, a n d o utso urci ng are all facets of a large mosaic in which incentives, co m munication and coordination, and the boundaries of the fir m are worked out. The outsourcing extends to the e mploy ment rela- tio ns hip, w here agai n t he relative costs of i n- house a nd t he outsourced re- so urces may have s hifte d. I haste n to a d d t hat t hese iss ues are far fro m bei ng settle d i n t he worl d of practice a n d i n t he worl d of eco no mic researc h.

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