Reveue Sharg ad Compettve Balace Does the varace proposto hold? Prof. dr. Stefa Kesee Ecoomcs Departmet Uversty of Atwerp ad Physcal Educato Departmet Catholc Uversty of Leuve (KUL) Uversty of Atwerp Prsstraat, 13, B-000 Atwerp Tel. 003-3-04105 Fa. 003-3-04799 e-mal: stefa.esee@ua.ac.be 1
Reveue Sharg ad Compettve Balace Does the varace proposto hold? Abstract Ths short paper, challegg the so-called varace proposto, argues that, for a geeral -team model wth proft mamzg clubs ad cocave reveue fuctos, there are reasos to beleve that reveue sharg ca worse the compettve balace amog teams a professoal sports league. If clubs are w mamzers, reveue sharg mproves the compettve balace.
Reveue Sharg ad Compettve Balace Does the varace proposto hold? 1. Itroducto Reveue sharg ad ts mpact o the compettve balace a sports league have bee dscussed by may sports ecoomsts durg the last decades. The so-called 'Ivarace proposto', clamg that reveue sharg does ot affect the dstrbuto of playg talet amog proft mamzg clubs, has bee challeged by may ecoomsts sce the semal artcle of Rotteberg (1956) ad the formal proof of the proposto by Qur ad El Hodr (1974). Atso, Staley ad Tschrhart (1988), Marburger (1997) ad Kesee (000) showed that, wth a more geeral club reveue fucto, that also cludes the absolute qualty of a team, or the wg percetage of the vstg team, reveue sharg ca mprove the compettve balace. Fort ad Qur (1995) showed that, f clubs have other reveues that are ot shared, reveue sharg ca worse the compettve balace. Kesee (001) showed that the effects of reveue sharg o compettve balace also deped o the specfcs of the sharg arragemet, such as match gate sharg or pool sharg. I all these studes, oe of the mportat mplct or eplct hypotheses of the model s that the supply of talet s costat, ad moreover that clubs tae ths fact to cosderato, meag that team maagers tae to accout that hrg a etra talet ot oly stregthe ther team, but also weaes aother team the same league, whch affects the margal reveue of talet. Ths hypothess ca be realstc, at least the short ru, for the closed North 3
Amerca Major Leagues, but t ca hardly be a realstc assumpto for the post- Bosma 1 Europea settg of may ope atoal soccer leagues. I a recet paper, Szymas ad Kesee (003) have show that reveue sharg worses the compettve balace f the supply of talet s ot costat, but fleble. Moreover, they argue that the costat supply approach Qur ad El-Hodr (1974), Fort ad Qur (1995) ad Vrooma (1995) s ot a Nash equlbrum, ad show how the Nash cocept mples a dfferet soluto. Also the costat-supply approach, reveue sharg worses the compettve balace. However, ther aalyss, the authors start from a smplfed two-team model, whle t has bee show that ot all results from a -club model, regardg the mpact of reveue sharg, apply to a -club model (Kesee, 000). I ths paper we try to fd out how reveue sharg affects the dstrbuto of playg talet amog the clubs a -club league f the supply of talet s costat or fleble. I secto, a approprate -club model s specfed. Secto 3 vestgates the mpact of a specfc reveue sharg arragemet ad secto 4 cocludes.. A smple -club model wth fleble talet supply Let's start from the followg cocave reveue fuctos for the clubs the league: R R( m, w ) wth 0 m > R w > 0 R < 0 w R w m > 0 for all (1) 1 The Bosma verdct (1995) by the Europea Court of Justce Luemburg establshed a free move of players betwee the coutres of the Europea Uo. 4
R s a club's total seaso reveue, w s the wg percetage ad m s the sze of the maret or the drawg potetal of the club. So, we assume that, besdes the sze of the maret, oly the ow wg percetage, or the relatve qualty of the team, affects seaso reveue. We further assume that the w percet of a team s equal to / tmes ts relatve playg stregth, so that we ca wrte that: w = for all () where s the umber of playg talets (ot the umber of players) of a team. The factor / s due to the fact of lfe that the sum of the wg percetages of teams does ot equal oe but /, whch s a costat, so that t ca be deleted wthout affectg the aalyss below. The sum of all talets the deomator of () ca be cosdered as the total supply of playg talet the league. So far, most aalyses, ths varable has bee, mplctly or eplctly, assumed to be costat, ad substtuted to the objectve fucto. I our approach, we assume the supply of talet to be fleble, gve a fed margal cost of talet. It s clear that the ope atoal leagues of Europea soccer, where clubs ca also hre talet from other atoal leagues, a etra talet for oe club does ot mply a loss of talet of aother club the same atoal league. It follows that the mpact of a etra talet has a smaller effect o club reveue, f that talet s hred form aother league, so that the margal reveue of talet ca be calculated as: 5
R R w R = = w w ( ) for all (3) The effect that a talet crease of oe club has o the reveue of aother club s the: R R w R = = w w ( ) j j j j j j j for all j (4) But, eve wth a costat supply of talet that s ot tae to accout by team maagers determg ther margal reveue of talet, the margal reveue of talet ca be wrtte as (3) ad (4). The dfferece wth the "teralsed" costat supply approach s clear: wth a fed talet supply equal oe, epressos (3) ad (4) would have bee:, whch ca be rescaled to R R 1 R = = w w Rj Rj wj 1 Rj 1 = = wj w wj 1 (5) (6) I ths costat supply case, t s perfectly possble, ad most lely, that the crease w percet of oe team oly affects the w percet of oe other team. However, for symmetry reasos, the last factor of (6) s based o the hypothess that all -1 teams bear a equal loss of w percet caused by the crease of the w percet of 6
oe team. Ths also mples that, a large league wth may teams, the rght had sde of (6) s close to zero. A smple -club model wth quadratc reveue fuctos shows the mpact of a fleble or a (o-teralzed) costat supply of talet o the talet dstrbuto amog clubs. Let the reveue fucto of a large ad a small maret club be: R = mw 0.5w R = mw 0.5w wth m > m (7) j j j j j The supply of talet s put equal to uty, ad s fully used a compettve (free agecy) player maret. Because proft mamzg clubs hre talet utl margal reveue equals margal cost, tag to accout the costat supply of talet, the talet dstrbuto s gve by: m w = mj w j so that: w 1 + m mj = = (8) w 1 + m m j j j If the costat supply s ot tae to cosderato, the dstrbuto s gve by: w m ( m w) j = ( mj wj) so that: = = wj j mj (9) Ths talet dstrbuto of talet (9) also emerges f the supply of talet s fleble, gve a margal cost of talet. Comparg the solutos (8) ad (9), t s also clear that, for ths smple model, the latter case results a more balaced competto tha 7
the former. The hypothess of a costat margal product of talet leads to a compettve domace bas. 3. The mpact of a reveue sharg arragemet The specfc sharg arragemet we start from ths chapter s a pool sharg system, whch cossts of the cotrbuto of a fed percetage of a club s total seaso reveue to a league fud that s redstrbuted- equally amog all clubs the league. The pool sharg system s dfferet from a typcal gate sharg system where home ad vstg team share the reveue of each match. The pool sharg system apples to the sharg of broadcastg rghts, sposorg ad merchadzg moey some Europea coutres ad the UEFA's Champos League. If µ s the share parameter ad ca wrte ths sharg arragemet as: * R s the club s total seaso reveue after sharg, we * (1 µ ) R = µr + R = µr + (1 µ ) R for all (10) whch ca also be wrtte as: * ( 1) µ 1 (1 µ ) R = + R + R for all (11) Note that ths pool sharg system ad the typcal gate sharg system are equvalet f there are oly two clubs the league. 8
To fd out what effect s of ths sharg system o the dstrbuto of playg talets the league, we wll ot use the approprate comparatve statc aalyss as Szymas ad Kesee (003), whch tured out to become very cumbersome, eve for a smple -team model. We used the approach of Marburger (1997) ad Kesee (000), vestgatg the shfts of the teams' demad curves for talet, caused by the sharg arragemet. Although ths approach, based o partal dervatves, does ot provde a full proof of the proposto, t does dcate that pool reveue sharg a -team model ca worse the compettve balace. The ratoale behd ths approach s the plausble assumpto that, f, for ay par of clubs the league, the dowward shft of the demad curve of the hgher taleted club s smaller tha the dowward shft of the demad curve of lower taleted club, measured at the tal player maret equlbrum pot, reveue sharg worses the compettve balace. If clubs are proft mamzers, the demad curve for playg talet s gve by the margal reveue product,.e.: R ( 1) µ + 1 R 1 µ R w w * = ( ( ) ( ) ) for all (1) I order to fd the shfts of these demad curves, the frst dervatve of (1) wth respect to the share parameter µ ca be calculated as: R ( 1) R 1 R = + ) > 0 for all (13) w w * / µ ( ( ) ( ) 9
Because ths epresso s clearly postve, reveue sharg causes all clubs to reduce ther demad for playg talet, for the obvous reasos that they all have to share ther margal reveue of hrg playg talet. More mportat for the purpose of our paper s the sze of these shfts of the demad curves at the tal equlbrum pot,.e. at the pot where the club's margal reveue equals the maret clearg margal cost of playg talet c before sharg,.e. where: R R w R w w = = ( ) = c for all (14) Substtutg (14) to (13) for all clubs yelds: * R ( 1) 1 / µ = c[ + ( j )] for all j j j (15) From ths result t ca be derved that the club wth the hgher umber of playg talets before sharg (whch s ot ecessarly the large-maret club), wll reduce ts demad for talet less tha the club wth the lower umber of playg talets,.e.: If > j the * R R j / µ < / µ for all j (16) j It follows that the reveue sharg arragemet worses the compettve balace. 10
Ths ca be llustrated wth a 3-club model. For the 3 clubs the dowward shfts of the demad curves, accordg to epresso (15), are: 1 3 For club 1: c [ + ( + )] 3 3 + + 1 3 1 1 1 3 For club : c [ + ( + )] 3 3 + + 3 1 1 1 For club 3: c [ + ( + )] 3 3 + + 3 1 3 (17) (18) (19) Comparg these shfts, t s clear that the epressos: (17) < (18) < (19) f 1 > > 3. (0) whch cofrms the geeral result (16) 3. A few remars - Epresso (15) also shows that, f the umber of teams the league s very hgh, the term bracets approaches oe, so that the dowward shft demad for talet of all teams s more or less the same. It follows that ths case the varace proposto stll stads. - A complcato of the model above s that, Europea soccer, clubs ca hre players from both ther ow atoal league ad from other foreg leagues. Ths m of playg talets a club ca be troduced the aalyss by assumg that the 3 It s also easy to chec wth ths method that the Qur ad El Hodr (1974) result was rght that reveue sharg does ot chage the compettve balace a closed league wth a fed talet supply. 11
playg talets our model () ca be decomposed atoal league talets 1 ad foreg league talets so that: = 1 + for all (1) The margal reveue of ow-league talet ad foreg-league talet are the gve by (5) ad (3) respectvely, wth (5) clearly larger tha (3). It follows that, assumg the same margal cost of ow-league talet ad foreg-league talet the Europea compettve player maret after Bosma, clubs wll rather hre talet from ther ow atoal league tha from foreg leagues, ceters parbus. The reaso s clear: whe a club hres ew playg talets from clubs ther ow atoal league, t reduces also the playg stregth of ts compettors the league, whch ehaces ts ow margal productvty. The pool reveue sharg arragemet (10) ths med-talet model wll stll worse the compettve balace, because of the uequal shft the demad for foreg talet by hgh taleted ad low taleted teams. Qur ad El Hodr (1974) have show that ths reveue sharg arragemet causes a equal dowward shft of the demad curves for ow-league talet. Ad, as show above, the club wth the better team reduces ts demad for foreg-league talet less tha the club wth the weaer team. The combato of these two results, a med-talet model, dcates that ths pool reveue sharg arragemet worses the dstrbuto of talet across teams a league. - Whereas the pool sharg system arragemet, as specfed (10), worses the compettve balace a league f all clubs are proft mamzers, t s obvous that * R 1 1 Usg (5) ad (6), result (15) becomes: / µ = c( + ) whch s the same for all. 1 1
reveue sharg mproves the compettve balace f clubs are w mamzers gve a fed proft rate, eve f the talet supply s fleble. I that scearo, the demad curve for talet s gve by the et average reveue curve (.e. after the subtracto of the fed captal cost ad the fed profts, whch ca also be zero). Sce the sharg arragemet (10) reduces the total reveue of every club that has a hgher reveue tha the average reveue the league, the result s a mproved compettve balace. - It s also possble that oe league, some clubs are proft mamzers ad others are w mamzers. I the most lely case of the poorer clubs beg w mamzers, ad the rcher ad more taleted clubs beg proft mamzers, the mpact of reveue sharg s clear. Because the poor clubs wll crease ther demad for talet ad the rch clubs wll reduce ther demad for talet, the mpact of reveue sharg s a better compettve balace. 4. Cocluso The mpact of reveue sharg o compettve balace s complcated, because t depeds o may factors, such as the objectves of the clubs, the specfcato of the reveue fuctos, the specfc sharg arragemet operato ad the supply of talet. Moreover, some results based o a smple two-team model do ot apply to a geeral -team model. Ths cotrbuto oly adds to our owledge that a pool reveue sharg arragemet a smple model wth a fleble supply of talet, whch s the case the atoal soccer leagues the Europea Uo, or a o-teralzed costat supply of talet, worses the compettve balace, at least, f clubs are proft mamzers. Ths result does ot mply that also gate reveue sharg worses the compettve balace a league, where also the w percet of the vstg team 13
affects club reveue, or where some club reveue s ot shared. For these complcatos, further research s eeded. 14
Refereces Atso S.E., Staley L.R., Tschrhart J., (1988), Reveue sharg as a cetve a agecy problem: a eample from the Natoal Football League, RAND Joural of Ecoomcs, vol. 19, r 1. Fort R., Qur J. (1995), Cross-subsdzato, Icetves ad outcomes Professoal Team Sports Leagues, Joural of Ecoomc Lterature, XXXIII, pp. 165-199 Kesee S., (000), Reveue Sharg ad Compettve Balace Professoal Team Sports, Joural of Sports Ecoomcs, vol. 1, r. 1, p.56-65 Kesee S., (001), The dfferet mpact of dfferet sharg systems o the compettve balace professoal team sports, Europea Sports Maagemet Quarterly, vol. 1. Nr. 3, p. 10-18. Marburger D.R., (1997), Gate Reveue Sharg ad Luury Taes Professoal Sports, Cotemporary Ecoomc Polcy, vol. XV, aprl Qur J., El Hodr M., (1974), The Ecoomc Theory of a Professoal Sports League, : Noll, R.G., ed. (1974), Govermet ad the Sport Busess, Broogs Isttuto, Washgto DC. Rotteberg Smo (1956), The Baseball Players' Labor Maret, Joural of Poltcal Ecoomy, 64 Szymas S. ad Kesee S. (003), Compettve balace ad gate reveue sharg team sports, Joural of Idustral Ecoomcs, forthcomg. Vrooma J., (1995) A geeral theory of professoal sports leagues, Souther Ecoomc Joural, 61 / 4 15