Career Lotto? Labor Supply in Winner-Take-All Markets

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1 Career Lotto? Labor Spply n Wnner-Take-All Markets Wayne A. Grove, Mchael Jetter and Kerry L. Papps Ho does the presence of sperstars, ho earn vast amonts, affect the partcpaton of more modestly-talented orkers n a nner-take-all labor market? We se longdnal data on nor and professonal tenns players to examne ho the decson to prse ths extremely rsky career s related to the earnngs a player s lkely to receve n the ftre. We fnd that teenage players prefer hgh mean earnngs and a lo varance, bt are also attracted to hghlyskeed earnngs dstrbtons st lke gamblers at horse races or n lotteres. The magndes of the skeness effects e estmate are mch smaller than n stdes of reglar gamblng behavor; hoever, they are stll szeable. If the skeness ere to fall to zero, boys old be arond 0% less lkely on average to contne n the sport, hereas grls old be 5% less lkely to contne. A sde-effect of nner-take-all labor markets therefore s an ncrease n partcpaton, even among those ho have a neglgble chance of earnng large amonts. 1. Introdcton The contempt of rsk and the presmptos hope of sccess are n no perod of lfe more actve than at the age at hch yong people choose ther professons ADAM SMITH 1776 Never gve p yor dreams s the nspratonal message ttered by Oscar nners, Olympc medalsts, aard-nnng athors, and others n arts, cltre, entertanment, and sports markets that offer mnscle probables of trly lfe-changng otcomes. 1 Does the presence of those and other sperstars, ho enoy fame and fortne, case sbstantal ndvdal and socal neffcency by encoragng modestly-talented orkers to prse long-shot careers h negatve expected vales, rather than more realstc occpatonal paths? Dobtng that sch asprants act as gddy rsk lovers h nrealstc assessments of themselves, Rosen and Sanderson 001 speclate that contnos feedback on one s performance cases entrants to sch to more realstc 1 For example, the 017 female nner of the Ne York Cy marathon, Shalane Flanagan, the frst Amercan n 40 years, sad: "I've been dreamng of a moment lke ths snce I as ltle grl Hopeflly nspres the next generaton of omen It took me seven years to do ths" Schonbrn, 017. Frank and Cook 1995, n The Wnner-Take-All Socety, most promnently advance the dea that the nner-take-all payoff strctre [n sperstar markets] generates a spral of ndvdal and socal occpatonal aste, snce leads both to ncreasng monetary and non-monetary reard neqales and to overcrodngs n the markets and occpatons prone to an overestmaton of one's chance to scceed. For example, the docmentary flm Hoop Dreams depcts the ltmately nsccessfl qest of to Afrcan-Amercan teenagers from poor Chcago neghborhoods to play for an NBA team. For example, a qarter of the parents hose chldren merely partcpate n hgh school sports hope ther chldren ll play n the pros, h mch hgher percentages for less edcated and less afflent parents NPR 015, p.17. 1

2 careers hen the prospects of makng the bg leages gets sffcently nfavorable. Ample evdence, thogh, contradcts sch expected tly theory predctons that ndvdals make decsons nder ncertanty based on the probables of otcomes. 3 For example, the sze of the ackpot, not the odds of nnng appears to drve lottery tcket sales. 4 Hence, e address the pzzle of hy so many people play career lotto by estmatng hether sperstars sccess nflences potental entrants to make large fnancal nvestments n exceedngly rsky, long-shot occpatons. 5 Alfred Marshall s 190 observaton almost a centry ago, pror to the dffson of broadcast and recordng technologes, that sngers salares ere constraned by the lmed nmber of persons ho can be reached by a hman voce antcpated the no bqos sperstar or nner-take-all markets; n sch markets, a very small nmber of performers satsfy most cstomers demands and earn the vast maory of the ncome. 6 Snce professonal ably n those markets only reveals self by on-the-ob talent dscovery, to be a contender, early aptde mst be nrtred and developed, hch may lead to years of skll-specfc hman capal nvestments; sch costs are estmated to exceed $100,000, for example, for tenns. 7 The Natonal Basketball Assocaton NBA llstrates the tny odds of sccess n sperstar labor markets: some 10,000 mpressve hgh school players of over 500,000 partcpants ve for abot 1,000 Dvson I freshman spots ho n trn compete to be one of abot 30 to on an NBA roster each year and earn over half a mllon dollars, of hom only abot half reman on the rooster for fve years. 8 In the past, some organzatons bore talent development rsks by sgnng exclsve long-rn labor contracts h a portfolo of 3 See the leratre on prospect theory Kahneman and Tversky 1979 and related salence models Dertnkel-Kalt and Koster See Clotfelter and Cook 1993 and the extensve feld and laboratory leratre on gamblng. 5 Accordng to Caves 000, one n 15,000 sbmted fcton manscrpts gets pblshed and then n 1994, for example, for athors Clancy, Crchton, Grsham and Kng acconted for 70 percent of the sales Sorensen Althogh Marshall 190 assmed that those reached by the hman voce old pay for the pleasre to do so, the pratng of recordngs and performances de to advances n dgalzaton ndermnes artsts ably to receve compensaton for ther performances. 7 See USTA and Lan Tenns Assocaton estmates. 8 See Rosen and Sanderson, 001. Reports by the Natonal Collegate Athletc Assocaton NCAA sggest extremely lo odds of college athletes playng on professonal football, baseball, and hockey teams. Althogh no comparable data on asprng orchestra members s provded, Chrstan Colberg, prncpal vola of the Cncnnat Symphony, sggests that from a statstcal chance, s probably easer to get nto the NBA.

3 asprng performers n, for example, the moton pctre stdo system for Hollyood actors and n professonal sports. 9 Snce the demse of exclsve long rn labor contracts, potental entrants to nner-take-all markets, or some backer, mst bear the development and annal opportny costs for a mnte chance of sperstardom. If sch career decsons really case sbstantal socal neffcency, hat market falres are to blame? We nvestgate to possble reasons for ndvdal mscalclatons of ther odds of sccess: 1 the lack of data abot the probables of earnng a lvng n a doman and decson-makng based on the desrably of lfechangng otcomes, rather than ther probably. 10 Althogh entrants to sperstar markets are often assmed to be rsk-lovers, mght ther rsk-takng really reflect attracton to posve skeness as shon by the consderable emprcal leratre of gamblng on horse races and lotteres? Gamblers ho accept a loer expected payoff n retrn for a small chance of a large gan are attracted to skeness, not to rsk. 11 In fact, evdence of the llngness to accept a loer expected payoff n retrn for greater skeness, rather than beng rsk lovng, seems to descrbe decson-makng n labor markets Hartog and Verberg 007 and regardng entreprenershp Chen et al. 016, savngs Goller 001, and fnancal nvestments Brnnermeer and Parker 005. Snce these leratres contan srprsngly lmed nformaton abot skeness attracton dfferences by gender, a contrbtons of ths paper s dong so for occpatonal choce decsons. What s mssng n the sperstar and nner-take-all markets leratre, and the most basc dlemma for the famles and yong asprng performers, s that data are not avalable to calclate meanngfl sccess probables for potental entrants 9 Before free agency, Maor Leage Baseball players sgned lfelong contracts h teams and n the 190s-1940s Hollyood actors sgned 7-year contracts h moton pctre companes Tervö Althogh most of the emprcal and expermental stdes cannot dstngsh beteen preferences for posve skeness verss mscalclatons abot the probably of sch otcomes, Snoberg and Wolfers 010 attempt to dstngsh beteen preference-based and percepton-based explanatons and fnd evdence that msperceptons of probables accont for the longshot bas among horse track gamblers. 11 Regardng the role of skeness n lottery and horse race gamblng, see Clotfelter and Cook 1993, fnd lotto sales to be posvely related to the sze of the ackpot, bt negatvely to expected vale. Golec and Tamarkn 1998, and Garrett and Sobel 1999, p.88 ho conclde that lottery players, lke horse race bettors, are rsk averse bt favor posve skeness. For the volmnos laboratory expermental leratre regardng the longshot anomaly and evdence for posve skeness preferences, see Grossman and Eckel

4 Rosen and Sanderson Whle yong volnsts, golfers, and opera sngers, for example, may focs on the spectaclar sccess of otler sperstars, the lack of nformaton abot the entre pool of nsccessfl contenders prevents systematcally calclatng the lkelhood a performer to earn a lvng n those markets, gven ther development traectory. Snce no obectve qaly measres exst for performers n the artstc, cltral, and entertanment markets, 13 Kreger 005, for example, calclates rock star qaly by contng the nmber of mllmeters of prnt colmns ncldng photos devoted to each artst n The Rollng Stone Encyclopeda of Rock & Roll p In hs semnal artcle The Economcs of Sperstars, Rosen 1981 asserts that confdentaly las and other dffcltes make vrtally mpossble to obtan systematc [earnngs] data n ths feld, referrng to classcal mscans, comedans and others n sho bsness. Many professonal sports offer soltons to those problems snce earnngs and prodctvy nformaton are pblcally avalable for at least some sports. 15 Beyond professonal earnngs and prodctvy data, hat has proven especally elsve for researchers has been obtanng pre-professonal obectve measres of ndvdal ably that relate to adlt sccess n the same doman, combned h professonal prodctvy and earnngs data. 16 Conseqently, very lmed emprcal analyses exsts of potental entrants probables or determnants of professonal sccess, even thogh the extensve leratres of talent development stdes Baker et al. 017, prodgy stdes Lbnsk et al. 014, expertse stdes Ercsson et al. 006, and ably selecton n sports and labor economcs are premsed on the early-n- 1 Althogh Rosen and Sanderson 001 refer to professonal sports, the same apples to the arts and other forms of entertanment as Caves 000 dscsses n Creatve Indstres. 13 See Menger 006. Althogh nmeros competons exst n the arts, cltral and entertanment markets, e kno of no analyss assessng ho predctve the nners of the yoth competons based on obectve dgments are of career otcomes n those domans. Exstng stdes that compare yoth to adlt otcomes typcally measre sccess as the achevement of an extraordnary accomplshment, sch as standng on an Olympc podm, becomng a top ranked player, or nnng a prestgos competon. 14 Entertanment s more abot fndng ot the tastes and hms of the pblc than abot the obectve measre of qaly Tervo 009, Hence, Kahn s 000 artcle entled: The Sports Bsness as a Labor Market Laboratory. Of corse, obectve prodctvy measres of athletes vary by sport, especally for team sports. And total earnngs may not be avalable snce endorsements and others forms of ncome besdes salares or prze nnngs are not necessarly pblcally avalable. 16 Kreger s 005 approach can offer no nformaton abot the labor spply decsons of potental entrants. Hamln 1991 and 1994 rated snger s voce qaly. 4

5 lfe observably of talent or of partclar characterstcs thoght to predct adlt otcomes. 17 Ths paper makes three key contrbtons. Frst, e contrbte to the leratre by assemblng the frst systematc and representatve longdnal dataset of potental entrants and of professonals n a sperstar market by lnkng obectve yoth rankngs of ables to adlt rankngs and earnngs n the same doman, from age 13 to 7 the length of almost all careers n the doman. 18 Importantly, the avalably of data for males and females perms s to nvestgate gender dfferences. We stdy tenns becase s the only doman, to or knoledge 19, hch offers global ordnal rankngs and rankng ponts for nor players 0 hch e lnk to professonal tenns crc tornament data and earnngs for omen and men, gathered from the ebses for the Women's Tenns Assocaton WTA and for the Assocaton of Tenns Professonals ATP, respectvely. Tenns offers obectve ably measres based on ndvdal performance otcomes n tenns matches and tornaments, hereas the sbectve dgements of scots determne the rankngs of athletes n team sports 1 and decsons by dges n artstc, cltral, and entertanment performers. Ths yoth/amater sample reasonably consttes a global pool of potental entrants to ths labor market. 17 Althogh there are some predctors of yoth characterstcs that predct general adlt otcomes, sch as the fact that the top 1 percent of mathematcal reasonng ably 13-year olds n the U.S. far exceeded average otcomes Lbnsk et al. 014, or nterest s abot doman-specfc otcomes and specfcally an actvy n hch yoth develop expertse and consder prsng a career n that doman. 18 Most stdes of sccess n nner-take-all markets analyze the determnants of achevng a goal, sch as nnng a prestgos aard, competon, or a spot on an Olympc team or an Olympc metal, rather than some varant of net lfetme earnngs. 19 Golf, the other maor lcratve ndvdal sport, lacks global nor rankng data. Chess offers better measres of ndvdal qaly, bt lacks sffcent compensaton to attract potental grandmasters from hgh ncome contres see Staray to Heaven. 0 The Internatonal Tenns Federaton began rankng age 18 and nder U18 players n The Tenns Erope Jnor Tor has pblshed age 14 and nder U14 and age 16 and nder U16 year-end rankngs snce 1990, hch ncldes players from arond the orld, althogh most are Eropean. 1 De to the dffclty of measrng ndvdal ables n team sports, yoth ably measres typcally constte the sbectve dgements of scots. Some lmed scotng data of yoths exst, bt have not yet been sed to predct professonal otcomes. For basketball, to scotng agences rank hgh school basketball players, ncldng freshmen to senors bt h many more older stdents, based on perceved qaly from see assocated h Yahoo! Sports and assocated h see Anderson and Snkey 013. Althogh baseball has long nclded hgh school stdents n scotng reports and have drafted hgh school gradates aged 17+, or analyss provdes ndcators of professonal sccess at mch earler ages. 5

6 For players as yong as age 13, ho have receved obectve sgnals abot ther relatve nternatonal performances and, hence, potental to earn a lvng on the professonal tenns tor, e estmate players lfetme earnngs dstrbtons from ther annal global rankngs. Based on those estmated career earnngs, e analyze players decsons to contne makng ncremental hman capal nvestments each year or to q and focs on other prss. The nner-take-all natre of professonal tenns gves rse to extremely skeed earnngs dstrbtons and e test hether ths skeness nflences players partcpaton decsons, along h the mean and varance of lfetme earnngs. We fnd that althogh teenage players prefer hgh mean earnngs and a lo varance, they are also attracted to hghly-skeed earnngs dstrbtons lke gamblers at horse races or n lotteres. If skeness ere to fall to zero, boys old be 3% less lkely on average to contne n the sport, hereas grls old be 5% less lkely to contne. A sde-effect of nner-take-all labor markets therefore s an ncrease n partcpaton, even among those ho have a neglgble chance of earnng large amonts. We offer to other maor contrbtons n addon to creatng the frst yoth-toprofessonal doman data set of ables and earnngs and ncorporatng skeness preferences nto occpatonal choce decson-makng. The frst of those s that e compare or analyss of hgh stakes career gambles and of lo stakes lotteres and fnd 0 tmes greater skeness preferences for lo stakes lottery players than for hgh stakes career lotto gambles. Fnally, or reslts reveal that boys exhb for tmes the skeness preference of grls n these exceedngly rsky occpatonal choces. The paper proceeds as follos. The next secton provdes backgrond nformaton abot decson-makng nder ncertanty. Then e develop a smple model that ncorporates the role of skeness n career choce. In secton 4 e ntrodce or and llstrate the relatonshps beteen lfetme earnngs and pre-professonal yoth rankngs. Secton 5 presents or man emprcal reslts and n secton 6 e compare the role of skeness attracton n or hgh stake career choce decsons compared to small stake lottery gambles, derved from Reger et al. And then e conclde. For an anecdotal accont, see Thosands of Players, Hndreds of Events and Ltle Reard, Davd Waldsten, Ne York Tmes, Agst 18,

7 . Backgrond To broad sets of theores have been proposed to explan decson-makng nder ncertanty. In a sketch of a standard ratonal expectatons expected tly model, Rosen and Sanderson 001 characterze nner-take-all labor market partcpants as dynamc learners abot ther ables and prospects of sccess, by reglarly reassessng the expected vale of ther lfetme earnngs or ther chances of ele employment, for example by a maor-cy orchestra or a professonal sports team. In a standard ratonal-expectatons expected-tly model, an agent eghts the sms of the tly vales of otcomes mltpled by ther respectve probables Von Nemann and Morgenstern 007. Sch a frameork s n the spr of the sperstar model by MacDonald 1988 n hch ably gradally reveals self over tme based on the accmlaton of nformaton abot one's performances h sperstar earnngs provdng the proper ncentves to enter these professons. Related ob-matchng models are provded by the sperstar model of Rosen 1986 and the occpatonal choce model of Mller Rosen and Sanderson 001 sggest that the opton vale of occpatonal rsk-takng encorages entry, bt also lms the rsk of socal and prvate losses, akn to the standard vale of an opton n fnance. Stange 01 estmates that opton vale acconts for 14 percent of the total vale of the opportny to attend college for the average hgh school gradate and s greatest for moderate-aptde stdents. Althogh expected tly theory assmes that all percentage ponts of rsk are eqally mportant, prospect theory proposes that the vales of otcomes of rsky prospects are mltpled by decson eghts that measre the mpact of events on the desrably of prospects, and not merely the perceved lkelhood of these events Kahneman and Tversky 1979, p. 80. As sggested by the Allas paradox, ample evdence ndcates that people vale very small changes n the probably of bg payoffs over larger changes nearer the mddle Savage 197, Slovc and Tversky Althogh entrants to sperstar markets are often assmed to be rsk lovers, the consderable emprcal leratre on gamblng on horse races and lotteres fnds a llngness to accept a loer expected payoff n retrn for greater skeness, rather than beng rsk lovng Golec and Tamarkn 1998, Clotfelter and Cook 1993, Garrett and Sobel The analogy h lotto s sefl n thnkng abot the dstrbton of ncomes as a matrx of payoffs. Clotfelter and Cook 1993 fnd lotto sales to be posvely related to the sze of the ackpot, bt negatvely to expected vale. Golec and 7

8 Tamarkn 1998 fnd that horse track bettors are attracted to the posve skeness of retrns offered by lo probably, hgh varance bets, rather than beng rsk lovers h mean-varance tly fnctons, as sggested by Qandt Garrett and Sobel 1999, p.88 conclde that lottery players, lke horse race bettors, are rsk averse bt favor posve skeness. 3 Althogh most of the emprcal and expermental stdes cannot dstngsh beteen preferences for posve skeness verss mscalclatons abot the probably of sch otcomes, Snoberg and Wolfers 010 attempt to dstngsh beteen preference-based and percepton-based explanatons and fnd evdence that msperceptons of probables accont for the longshot bas among horse track gamblers. Whereas lotto and lottery games ental prely random otcomes, e analyze nner-take-all labor market reslts based on ndvdal ables. What has remaned elsve for nderstandng labor spply n nner-take-all markets are longdnal datasets that lnk yoth and adlt relatve performances to career earnngs. Related emprcal analyses of decsons made n the presence of ncertanty abot one's on ables and expected ftre otcomes nvestgate hether or not to attend college Kane 1994, f n college, hether to contne or to dropot Stange 01, Stnebrckner and Stnebrckner 01, and the choce of a college maor Arcdacono et al. 01. Generally, these athors fnd a reasonable approxmaton to Bayesan pdatng based on seqental experences that provde ne nformaton abot ndvdals match h partclar tranng programs. 3. Model Sperstar markets are bnary snce, for example, drafted players and other sperstars earn n the NBA and non-drafted players earn nothng and typcally do not have ancllary careers, sch as coachng, that offer lfetme ncomes comparable to that of a college edcated orker. Althogh an ele grop of sperstars, for example, accontants, layers or dentsts ho earn standard devatons greater than others, those professons do no constte nner-take-most markets n or ve, as Frank and Cook 3 For the volmnos laboratory expermental leratre regardng the longshot anomaly and evdence for posve skeness preferences, see Grossman and Eckel

9 9 010 sggest, becase the typcal professonal n those felds earns a posve rate of retrn on ther hman capal nvestment. Assme that people have a choce beteen a rsky career, n hch ther ftre earnngs are ncertan, and a rskless career, hch pays a gven amont h certanty. People ork as apprentces for T perods before ther fnal earnngs,, are revealed. Wages drng the apprentceshp phase are assmed to be the same on the rsky and rskless ob. Fnal earnngs each perod on the rsky ob are determned by a person s ordnal rankng accordng to performance, hch s not knon n advance. Hoever, at the end of each perod t, people learn ther crrent rankng, r. In each apprentceshp perod t, person ll choose to contne h the rsky career f hs/her expected earnngs n the post-apprentceshp perod are greater than on the rskless career, that s: t E, here s the person s tly fncton over lfetme ncome, s lfetme ncome on the rsky career, s lfetme ncome on the rskless career. The person s tly fncton can be approxmated by a Taylor seres expanson arond the mean of,, Golec and Tamarkn 1998: Then, takng expectatons at tme t and addng an error term: t t t t E E E E 3 6. Let ], [ ~ e e U. Then the probably that the person ll contne on the rsky career after perod t s: ˆ rsky career E E P P t t 6 3 E t 4 ˆ E e E e e e t t 3 1 E e t.

10 If the person s rsk netral, 0 ; f the person s skeness netral, 0. Hence, estmates of a person s sqared and cbed devatons of lfetme earnngs can be added to an eqaton for the probably of contnng n the rsky career. A sgnfcant posve coeffcent on the cbed term ndcates that the person s skeness lovng. In each perod of the apprentceshp phase, a person s expectatons depend on hs/her rankngs p to that pont. Therefore, eqaton 1 can be rerten as follos: e ˆ Prsky career E e e r E 4e 3 E r, 3 1e here r s the seqence of person s rankngs p to perod t. r 4. Data Or dataset combnes for dstnct sorces for tenns rankngs and earnngs, all of hch are avalable for females and males. Overall, e nclde every player born beteen 1977 and 1986 ho appeared at least once n an nternatonal rankng beteen the ages of 13 and 30. Frst, snce s ncepton n 1990, the Tenns Erope Jnor Tor pblshes year-end rankngs for U14 and U16 players from arond the globe ho compete n nmeros tornaments throghot Erope. These data provde each player s fll name, contry of orgn, brthday, and rankng. Ths Tor represents the earlest and most comprehensve rankngs avalable for tenns players competng n nternatonal events. Second, e access the U18 orldde rankngs, pblshed by the Internatonal Tenns Federaton ITF. For a yong potental entrant nto professonal tenns, ths Tor provdes the next and fnal step before enterng the professonal arena. Smlar to Tenns Erope, the ITF data nclde each player s fll name, brth date, natonaly, and rankng. Thrd and fnal, data regardng players professonal performances comes from the respectve professonal organzatons: the Women's Tenns Assocaton WTA and the Assocaton of Tenns Professonals for men ATP. To derve a comparable nverse of players throghot all age and rankng categores over tme, or analyss focses on the cohort of players born beteen 1977 and Ths tmeframe garantees observng players performances from age 13 hen Tenns Erope rankngs became avalable to 30 snce e record players' professonal performance 10

11 ntl the end of 016. Or dataset ncldes any player born n that age cohort ho shos p at least once n any of the above categores of rankngs. In hs reve of The Wnner-Take-All Socety by Frank and Cook 1995, Rosen 1996 arges that fe serosly try to enter these professons... and those that do are predomnantly nsklled and trobled people ho never played the game at all so that the the neffcences they [Frank and Cook] clam seem to me be greatly exaggerated 134. Fgre 1 plots a player s annal prze money n the WTA or ATP aganst hs/her year-end rankng. Ths llstrates ho closely related players earnngs are to ther rankng, as ell as the nner-take-all natre of professonal tenns. Becase prze money s allocated accordng to a player s poson n each tornament and becase ths declnes sharply h tornament poson, the top male and female players accont for the vast maory of total prze money. The decson facng yong players s hether to rsk enterng the professonal tenns labor market or to choose a safer career. To do so, they mst evalate and compare the expected dstrbtons of lfetme earnngs for the to careers. If players care only abot ther expected lfetme earnngs, the amont of labor sppled shold match closely the expected earnngs. Hoever, hle lfetme prze money s hghly non-lnearly related to nor rankng as seen n the bottom panel of Fgre, a person s lfetme nmber of tornaments s roghly lnearly related to rankng as seen n the top panel. Or analyss reqres estmates of the lfetme prze money dstrbton a gven player expects to face. In each year, e calclate a player s rankng hn the cohort of players of the same sex born n the same year, b, n eher the U14, U16 or U18 age categores or the professonal tor. 4 A player s total ftre prze money, W, s determned by hs/her age-specfc rankng n each ftre year, r: b 30 v1 W W r r. 4 tv We assme that players only receve prze money beteen ages 19 and 30. Snce players do not kno the ftre prze money dstrbton, e assme that they se the 4 For players ho have a rankng n more than one age category n a year e.g. a 16 year-old ho plays n both the U16 and U18 categores, the best age-specfc rankng s taken. 11

12 dstrbton of prze money on the professonal tor.e. WTA and ATP n the prevos year as a reference pont: r tv rt v, a v, s, t 1, 5 here the rght-hand sde gves the prze money of a player ranked r among a cohort of players aged a and of sex s n year t 1. To smplfy the comptaton, prze money s ronded to the nearest $1,000. The PDF for the lfetme prze money dstrbton gves the probably of each lfetme earnngs amont arsng, hch s eqal to the probably of a gven seqence of rankngs over a player s lfetme: P W P r, r,..., r. 6 t30a t9a t1 Players cannot possbly kno the probably of a gven seqence of rankngs, snce there are contless sch seqences. Hoever, seems reasonable to assme that players old kno ho the probably of any gven rankng n the follong year s related to ther crrent age-specfc rankng. If e assme that the probably of any rankng s determned solely by a player s age-specfc rankng n the prevos year, e can smplfy the prevos expresson as follos: P W P rt v rt v, a v 1, s P rt v1 rt v, a v, s... P rt 1 r, a 1, s. 7 The year-to-year transon probables are calclated by comparng the rankngs of all players of a gven age and sex n one year and the next. To smplfy the comptaton, e grop players nto 10-rankng bands beteen rankngs 50 and 500 and 100-rankng bands above rankng 500. Combnng the age-specfc transon probables and the age-specfc earnngs dstrbtons and consderng every possble rank n every ftre year of a player s career, e can calclate lfetme prze money PDFs for each player n each year. These exhb sgnfcant varaton. Fgre 3 plots the dstrbtons faced by 18-year olds n 1997 near the mddle of or sample h dfferent rankngs. Even those top- ranked n ther age grop face a reasonably hgh chance of earnng very ltle over ther careers, especally boys. Hoever, those ranked 100 experence a mch hgher lkelhood that players ll earn close to zero over ther careers. Ths fact s reflected n the skeness 3 coeffcents for dstrbtons calclated as E W W E W W and reported nder each hstogram, hch are mch hgher for those ranked 100 than for those ranked

13 As players age, ther rankngs become better and better predctors of ther lfetme earnngs. Fgres 4 and 5 plot the lfetme prze money dstrbtons for players ranked frst among ther cohort at each age beteen 13 and 18. Top-ranked 13-year-olds have a very hgh chance of makng ltle money over ther careers, compared to top-ranked 18-year-olds. Accordngly, the skeness coeffcent falls h age. We can calclate three moments of the lfetme earnngs dstrbtons for any player n any year t: E W P W W ; W E W E W W E W ; W 3 3 E W E W W E W. 8 W A player s consdered to be actve n tenns n a gven year f he/she partcpated n any of the tenns dvsons n or dataset U14, U16, U18 or the professonal tors. The means for the prmary varables of nterest are reported n Table 1. Mean expected prze money s slghtly hgher for grls than for boys, even thogh professonal men are pad more than eqal-ranked omen, becase more boys partcpate n nor tenns and they face a hgher chance of earnng very ltle over ther careers. Boys also have a mch greater expected varance and skeness than grls. 5. Estmates of the model of tenns partcpaton A dmmy varable for hether a player as actve s regressed on the mean, varance and skeness of hs/her expected career prze money, gven hs/her rankng n the prevos year, consstent h eqaton 3, as follos: P actve W 1 Et 1 W Et1 W AGE β, Et 1 W W here AGE s a fll set of age dmmes, ntended to captre the effects of changes n opportny cost over a person s teenage years, γ s a player fxed effect hch s omted n some specfcatons and ε s a random error term. The reslts for boys are reported n Panel A of Table. As predcted by theory, the estmates of 1, and 3 are posve, negatve and posve, respectvely. In the frst colmn, the coeffcents mply an elastcy h respect to the mean of prze money of 0.610, an elastcy h respect to the varance of and an elastcy h respect 13

14 to the skeness of 0.8. Ths means that f the skeness of the prze money dstrbton ere to fall to zero, hot a change n the mean and varance, the average boy old be 3 percent less lkely to contne playng tenns the follong year. The age coeffcents ndcate that players are ncreasngly lkely to q tenns as they age, h an especally large fall n partcpaton at age 19, hen most players have to decde hether to attend college. The reslts for grls are reported n Panel B of Table. The coeffcents n the frst colmn mply an elastcy h respect to the mean of prze money of 0.76, an elastcy h respect to the varance of and an elastcy h respect to the skeness of 0.045, meanng that grls old be 5 percent less lkely to stay n tenns on average f skeness fell to zero. The reslts n the frst colmn of Table do not take nto accont the cost of competng n professonal tenns. Snce top-ranked players travel more dely and reqre more spport staff than loer-ranked players, or estmates may overstate the skeness n lfetme earnngs. To address ths, e sbtracted an estmate of costadsted prze money, provded to asprng players by the Internatonal Tenns Federaton. Ths vares by sex, contnent and rankng. 5 As seen n the second colmn of Table, adstng for costs makes ltle dfference to the reslts. The skeness elastces fall by the same magnde for boys and grls, bt loses sgnfcance for grls gven the small amont of skeness affecton exhbed by year olds Table, panel B, colmn 1. In the thrd colmn of Table, e repeat the specfcaton n the frst colmn, sng observatons for ages 0-8. These players are already n the professonal tor and are consderng hether to contne or q, based on ther crrent rankng. We fnd mch smaller skeness effects and the coeffcent for omen s no nsgnfcant. Over the fll 14-8 age range, hoever, both men and omen exhb skeness lovng behavor as seen n the last colmn of Table. 6. Heterogeney n skeness preferences by sze of gamble 5 Snce e have no nformaton on a player s crrent place of resdence, e se ther contnent of brth. For players ho ere mssng ths nformaton, e se the costs faced by those lvng n Erope, snce ths s the most common contnent of brth. 14

15 Whereas lotto and lottery games ental small stakes that are prely random otcomes, e analyze nner-take-all labor market reslts based on ndvdal ables. Ho mght one expect skeness preferences to compare for lotteres and career lotto? Althogh a person mght prchase a negatve expected retrn lottery tcket for $ for the entertanment vale of the fantasy of htng the ackpot, career lotto choces may alter or fndamentally lm a person s occpatonal opportny set. Nonetheless, despe the stakes beng mch hgher, the reslts dscssed above sggest that ndvdals approach the choce of career n a manner remnscent of ho they approach lotteres. To examne ho or estmates of the magnde of the skeness effect compare h those exhbed n smaller lotteres, e compare or reslts h those obtaned from the expermental data collected by Reger et al Reger et al. asked partcpants n laboratory experments n college classrooms n 5 contres to gve certanty eqvalents for a seres of hypothetcal lotteres n hch they stood to gan at most $10,000 or to lose at most $100, h an average payoff of arond $800. Althogh the athors dd not examne attdes toards skeness, estmates of the relatonshp beteen the certanty eqvalent and the skeness can nonetheless be derved from ther data. 6 Each lottery has a dfferent mean, varance and skeness, therefore e calclate standardzed lotteres by sbtractng the mean payot from each lottery and dvdng by the standard devaton of the payots, so that the payots have mean zero and varance one n each case. In general, person s rsk premm for lottery can be rten as follos: 3 y E E, 10 6 here y s the person s certanty eqvalent, s a possble payoff for the lottery and s the mean payoff. Therefore, the rsk premm for standardzed lottery can be rten: E y 0 0 E 0 60 E These are avalable at 15

16 Eqaton 11 mples that the standardzed rsk prema are lnearly related to the skeness of a lottery. Therefore, e estmated ths relatonshp sng OLS, allong for person fxed effects. For comparably h or tenns estmates, e sed only those aged althogh ths made ltle dfference to the reslts. 7 The slope coeffcents provde estmates of a person s llngness to pay for a n of skeness, or skeness preference hch ll be negatve f people are skeness lovng, and the ntercept provdes an estmate of a person s rsk preference, specfcally the Arro-Pratt measre of rsk averson dvded by to hch ll be posve f people are rsk averse. Comparng eqaton 3 h eqaton 11, s clear that the rato of the coeffcent on the skeness and coeffcent on the mean n the former are comparable h the coeffcent on skeness n the latter. Both provde an estmate of a person s preference for skeness, relatve to ther preference for the mean that s, the thrd dervatve of ther tly fncton, dvded by sx tmes the frst dervatve. Hoever, snce the career lottery n or tenns stdy s over sch a vastly greater amont than n the expermental stdes an average lfetme payoff of arond $300,000, compared to $800 n Reger et al., the tly fnctons have been lnearzed arond dfferent ponts n the doman and the to sets of estmates provde an ndcaton of ho mportant skeness s to people hen assessng gambles of dfferent magndes. As reported n Table 3, the Reger et al. data reveal a relatve skeness preference for boys of and a relatve skeness preference for grls of 0.473, hch are 1 and 14 tmes larger than the correspondng estmates from the tenns career decson, respectvely. 8 Hence, skeness preferences have mch larger effects on the decson to prchase a lottery tcket than on the decson to enter a rsky career. Ths ndcates that people focs on the mean and varance of the potental otcomes hen a decson s lfe changng and are less nflenced by long-shot otcomes even thogh a preference for lo-probably, hgh-retrn otcomes stll has a szeable effect on behavor, as seen n Table. For both datasets, e also combned observatons for both genders and alloed nteractons h age and the GDP qntle of a person s contry of brth. Relatve skeness preference declnes slghtly h age n both datasets. Althogh there s a 7 Reger et al. dd not nclde anyone nder 16 n ther stdy. 8 Note that althogh e fond n Table that the overall skeness elastcy as smaller for grls than for boys, snce grls ere also mch less sensve to the mean, ther relatve skeness preference s larger than boys, as also fond n the Reger et al. data. 16

17 monotonc relatonshp beteen GDP and relatve skeness preference over modestszed lotteres, e fnd a U-shaped relatonshp n the career choce settng. Yong players from the poorest contres are relatvely attracted to hghly-skeed earnngs dstrbtons, nlke hat e fond h the Reger et al. data. 7. Conclson The fndamental dlemma h labor spply decsons n nner-take-all markets s that the beaty, fame and fortne of sperstars entces potental entrants to devote sbstantal pfront skll-specfc hman capal nvestments hot data to allo performers, parents and coaches to calclate meanngfl sccess probables. The pblc polcy concern regardng asprng partcpants n these markets s the sgnfcant ndvdal and socal neffcency that reslts from the resorces spent developng sklls for lo probably, hgh payoff careers, rather than nvestng n more realstc occpatonal paths. Ample behavoral economcs analyses ndcate the dffclty yong agents asprng performers and emotonally-nvolved ones.e. parents have fnctonng as ratonal Bayesan pdaters hen makng ncertan ntertemporal career decsons. Here e address the pzzle of hy so many people play career lotto by makng large fnancal occpaton choce gambles n exceedngly rsky, long-shot sperstar markets. Sch labor markets are characterzed by extremely skeed earnngs dstrbtons. We offer the frst emprcal analyss of labor spply decsons nder rsk n nner-take-all markets by creatng a nqe longdnal dataset h obectve global rankngs of yoth and of professonal partcpants and of career earnngs. Or data perm estmaton of the ratonal expectatons expected tly model sggested by Rosen and Sanderson 001: do nner-take-all labor market partcpants act as dynamc learners abot ther ables and prospects of sccess, reglarly reassessng the expected vale of ther lfetme earnngs or ther chances of ele employment, for example by a maor-cy orchestra or a professonal sports team, and ex hen the prospects become too lo? Usng longdnal data on asprng and actal professonal tenns players, e estmate a lfetme prze money dstrbton for each player, gven hs/her age, crrent rankng and the observed dstrbton of prze money among crrent professonals. We fnd that teenagers are more lkely to stay n tenns f they face a dstrbton h a hgh mean, lo varance and hgh skeness. All else eqal, boys old be 3% less and 17

18 grls 5% less lkely to contne n tenns each year f the skeness of each player s lfetme prze money dstrbton ere redced to zero. The fact that players are attracted to hghly-skeed dstrbtons s remnscent of the behavor of gamblers at horse races or of lottery entrants. Comparng or reslts h expermental data on modest-szed lotteres, e fnd that people exhb a mch smaller preference for skeness hen choosng careers. Nonetheless, the magnde of or estmates sggests that a love of skeness, among boys especally, can sgnfcantly ncrease labor spply n nner-take-all markets. References Anderson, B. C., and M. J. Snkey 013. Lke Mke or lke LeBron: Do the most able need college to sgnal? Workng paper. Arcdacono, P., V. J. Hotz, and S. Kang 01. Modelng college maor choces sng elced measres of expectatons and conterfactals. Jornal of Econometrcs 166 1, Baker, J., S. Cobley, J. Schorer, and N. Watte 017. Rotledge Handbook of Talent Identfcaton and Development n Sport. Abngdon: Rotledge. Clotfelter, C. T., and P. J. Cook The peclar scale economes of lotto. Amercan Economc Reve 83 3, Dertnkel-Kalt, M., and M. Koster 017. Salent compromses n the nesvendor game. Jornal of Economc Behavor and Organzaton, forthcomng. Ercsson, K. A., N. Charness, P. J. Feltovch, and R. R. Hoffman, eds 006. The Cambrdge Handbook of Expertse and Expert Performance. Cambrdge: Cambrdge Unversy Press. Frank, R. H., and P. J. Cook The nner-take-all socety: Why the fe at the top get so mch more than the rest of s. Ne York: Pengn. 18

19 Garrett, T. A,. and R. S. Sobel Gamblers favor skeness, not rsk: Frther evdence from Uned States' lottery games. Economcs Letters 63 1, Golec, J., and M. Tamarkn Bettors love skeness, not rsk, at the horse track. Jornal of Polcal Economy 106 1, Grossman, P. J., and C. C. Eckel 015. Lovng the long shot: Rsk takng h skeed lotteres. Jornal of Rsk and Uncertanty 51 3, Kahneman, D., and A. Tversky Prospect theory: An analyss of decson nder rsk. Econometrca 47, Kahn, L. M The sports bsness as a labor market laboratory. Jornal of Economc Perspectves 14 3, Kane, T. J College entry by blacks snce 1970: The role of college costs, famly backgrond, and the retrns to edcaton. Jornal of Polcal Economy 10 5, Kreger, A The economcs of real sperstars: The market for rock concerts n the materal orld. Jornal of Labor Economcs 3 1, Lbnsk, D., C. P. Benbo, and H. J. Kell 014. Lfe paths and accomplshments of mathematcally precocos males and females for decades later. Psychologcal Scence 5 1, MacDonald, G The economcs of rsng stars. Amercan Economc Reve 78 1, Marshall, A Prncples of economcs: Unabrdged eghth edon. Ne York: Cosmo Classcs. Mller, R. A Job matchng and occpatonal choce. Jornal of Polcal Economy 9 6,

20 NPR 015. Sports and health n Amerca. Accessed: Qandt, R. E Bettng and eqlbrm. Qarterly Jornal of Economcs 101 1, Reger, M. O., M. Wang, and T. Hens Rsk preferences arond the orld. Management Scence 61 3, Rosen, S The economcs of sperstars. Amercan Economc Reve 71 5, Rosen, S The theory of eqalzng dfferences. Handbook of Labor Economcs 1, Rosen, S., and A. Sanderson 001. Labor markets n professonal sports. Economc Jornal , Savage, L. J The fondatons of statstcs. Ne York: Dover Pblcatons. Schonbrn, S Shalane Flanagan Solves N.Y.C. Marathon for Amercan Women, Ne York Tmes, November 5, F1. Slovc, P., and A. Tversky Who accepts Savage s axom? Behavoral Scence 19 6, Smh, A An Inqry nto the Natre and Cases of the Wealth of Natons. Ne York: Modern Lbrary. Snoberg, E., and J. Wolfers 010. Explanng the favore-longshot bas: Is rsklove or msperceptons? Jornal of Polcal Economy 118 4,

21 Stange, K. M. 01. An emprcal nvestgaton of the opton vale of college enrollment. Amercan Economc Jornal: Appled Economcs 4 1, Stnebrckner, T., and R. Stnebrckner 01. Learnng abot academc ably and the college dropot decson. Jornal of Labor Economcs 30 4, Tervö, M Sperstars and medocres: Market falre n the dscovery of talent. Reve of Economc Stdes 7, Von Nemann, J., and O. Morgenstern 007. Theory of games and economc behavor. Prnceton: Prnceton Unversy Press. 1

22 Fgre 1 Year-end professonal rankng and prze money n 1997

23 Fgre Lfetme nmber of tornaments and average prze money by rankng at age 18 3

24 Fgre 3 Hstograms of predcted career prze money for 18-year-olds n 1997 Note: The dotted lne denotes the maxmm career prze money. 4

25 Fgre 4 Hstograms of predcted career prze money for boys n 1997 Note: The dotted lne denotes the maxmm career prze money. 5

26 Fgre 5 Hstograms of predcted career prze money for grls n 1997 Note: The dotted lne denotes the maxmm career prze money. 6

27 Table 1 Means of the key varables Varable Boys Grls Actve Lagged mean of predcted career prze money Lagged varance of predcted career prze money Lagged skeness of predcted career prze money Lagged rankng Age Nmber of observatons 15,810 15,078 Note: The mean of career prze money s expressed n mllons, the varance n trllons and the skeness s n qntllons of 010 US dollars. 7

28 Table Reslts of estmatng partcpaton eqaton A. Boys Varable Ages Ages 0-8 Ages 14-8 v Lagged mean of career prze money mllons of 010 US dollars 1.16*** [0.610] 0.904*** [0.179] 0.119*** [0.06] 0.160*** [0.083] Lagged varance of career prze money trllons of 010 US dollars Lagged skeness of career prze money qntllons of 010 US dollars *** [-0.554] *** [0.8] *** [-0.386] *** [0.179] *** [-0.050] ** [0.014] *** [-0.076] *** [0.063] R-sqared Nmber of observatons 15,810 15,810 6,009 1,819 B. Grls Varable Ages Ages 0-8 Ages 14-8 v Lagged mean of career prze money mllons of 010 US dollars 0.608*** [0.76] 0.37*** [-0.017] 0.46*** [0.098] 0.37*** [0.163] Lagged varance of career prze money trllons of 010 US dollars Lagged skeness of career prze money qntllons of 010 US dollars *** [-0.146] *** [0.045] [-0.009] [-0.001] *** [-0.046] [0.001] *** [-0.087] *** [0.033] 0.00 R-sqared Nmber of observatons 15,078 15,078 4,693 19,771 Notes: All specfcatons nclde a fll set of age and player fxed effects. In the second colmn, prze money s adsted for the estmated costs of competng. Elastces at the mean are presented n brackets. Standard errors are presented n parentheses. *, ** and *** denote sgnfcance at the 10%, 5% and 1% level, respectvely. 8

29 Table 3 Relatve skeness preferences by demographc grop Poplaton grop Gamble Contne tenns Reger et al. lotteres Men 0.019*** 0.404*** Women 0.033*** 0.473*** Age *** 0.465*** Age *** Age *** 0.441*** Age *** 0.440*** Brth contry GDP qntle *** 0.165*** Brth contry GDP qntle 0.09*** 0.73*** Brth contry GDP qntle * 0.401*** Brth contry GDP qntle *** 0.47*** Brth contry GDP qntle *** 0.438*** Notes: Vales n the frst colmn are ratos of the coeffcents on the skeness of predcted prze money and the mean of predcted prze money from a regresson of eqaton 9 sng ages Vales n the second colmn are coeffcents on the skeness of a lottery from a regresson of eqaton 11. *, ** and *** denote sgnfcance at the 10%, 5% and 1% level, respectvely. 9