MPRA
Munich Personal RePEc Archive
A NLIP Model on Wage Dispersion and
Team Performance
Papahristodoulou, Christos
M¨alardalen University/Industrial Economics
30. May 2012
Online at
http://mpra.ub.uni-muenchen.de/39149/
1
A N LIP Mod el on Wage Disp ersion and Team Perform ance
Christos Pap ahristod ou lou *
A bstract
Using a N on-Linear Integer Program m ing (N LIP) m od el, I exam ine if w age d ifferences betw een Su p er talents and N orm al p layers im prove the p erform ance of fou r team s w hich p articip ate in a tou rnam ent, su ch as in the UEFA Cham p ions Leagu e (UCL) grou p m atches. With ad -hoc w age d ifferences, the op tim al solutions of the m od el show that higher w age equ ality seem s to im p rove the p erform ance of all team s, irresp ectively if the elasticity of su bstitu tion betw een Sup er- and N orm al- p layers is high or low . In ad d ition to that, a U -typ e p erform ance exists in tw o team s w ith the highest and the second high elasticity of su bstitu tion. With team d ata from the 2011-12 UCL grou p m atches and from the Italian Serie A over 2010-12 seasons, the w age d isp ersion has no effect on team p erform ances.
Keyw ord s: Players, Team s, Wages, Perform ance, Tou rnam ent
*Dep artm ent of Ind u strial Econom ics & Organization, Mälard alen University , Västerås, Sw ed en; christos.p ap ahristod ou lou @m d h.se Tel; +4621543176
2
1. Introduction
The op tim al w age stru ctu re of team s has been frequ ently d iscu ssed over the last d ecad es. The m ain hyp othesis is w hether the com p ressed or the d isp ersed w ages among a team’s players have a stronger impact on the p erformance of the team. As is very often in econom ics, there are at least tw o schools of thou ght. Lazear & Rosen (1981) argu ed that the team p erform ance in creases if the best talents are p aid higher w ages than the norm al p layers. Milgrom (1988) and Lazear (1989) on the other hand , stressed the p ossibility of bad field coop eration betw een p layers, and consequ ently the p erform ance cou ld be inferior, if the u n d er-p aid p layers feel d iscrim inated . Levine (1991) took an extrem e position and favou red the egalitarian w ages. Fehr and Schm id t (1999) tried to balance these tw o effects and argu ed that, as a w hole, the team losses m ore if w ages are m ore u nequ al than equ a l.
Franck & N ü esch (2007) review ed the em p irical stu d ies from sp ort team s, in baseball, hockey, basket and football. Som e stu d ies seem to su p p ort the com p ressed w ages hyp othesis. They argu e that these find ings can p artly be exp lained becau se the m ajority of em p irical stu d ies assu m e a linear relationship betw een w age d isp ersion and team p erform ance. In their ow n stu d y, based on 5281 ind ivid u al salary p roxies from Germ an soccer players betw een 1995-07, they allow ed for squ ares of the Gini coefficient and th e coefficient of variation of the w age d istribu tion a t the beginning of the season. They fou nd a U-form ed sp ortive su ccess, i.e. team s p erform better by either an egalitarian pay stru ctu re or a steep one1. On the other hand , Pokorny (2004) fou nd an inverted U-form ed su ccess, i.e. the p erform ance is higher w ith interm ed iate w age d ifferences. Avru tin & Som m ers (2007), u sing baseball d ata from 2001-05, fou nd no effect. Torgler et al (2008), u sing also d ata from the Germ an Bu nd esliga (and the N BA), fou nd that p layers care m ore abou t the salary d istribu tion w ithin the team and not ju st abou t their ow n salary. Generally, p layers p refer a red u ced inequ ality and in that case their p erform ance im p roves. In ad d ition, a d etailed investigation of the basketball d ata sho w s also that w hen a p layer m oves from a relative incom e ad vantage to a relative d isad vantage, his p erform ance d ecreases in a statistically significant w ay. On the other hand , m oving from relative incom e d isad vantage to relative ad vantage has no effects. Wisem an & Chatterjee (2003), u sing baseball d ata from 1980-02, fou nd a negative effect of w age d isp ersion. In a sim ilar stu d y recently, i.e. w ith baseball d ata from 1985-10, Breu nig et al (2012) fou nd also a negative effect of w age d isp ersion. On the other h and , Sim m ons & Berri (2011), u sing basket statistics, su p p orted the Lazear & Rosen (1981) hyp othesis that higher w age d isp ersion increases team p erform ance.
1 Also they found evidence that teams with dispersed wages entertain the public better since the
3
In this p ap er I form u late and solve a N LIP m od el to exam ine m ainly the effects of w age d ifferences betw een Su p er (S) - and N orm al (N ) - p layers in the p erform ance of fou r team s, w hich p articip ate in a tournam ent like the UEFA Cham p ions Leagu e (UCL) grou p m atches. Using ad -hoc w age d ifferences and variou s “team p rod u ction functions”, it is of interest to see if the optimal solutions favour the compressed or the d isp ersed w ages hyp otheses for all, or som e team s. In ad d ition, tw o sm all d ata sets (UCL grou p m atches from 2011-12 and Italian Serie A from 2010-12 seasons) are u sed to test the above hyp otheses.
2.1 The model
There are fou r football team s w hich p lay three hom e and three aw ay m atches in the grou p stage of the UCL tou rnam ent. As a w hole there are 12 m atches and the m axim u m nu m ber of p oints is 36. The tw o team s w hich collect m ost p oints are qu alified for the next rou nd and the other tw o are elim inated2.
The fou r p articip ating team s have d ifferent qu alities and consequ ently d ifferent ranking. To differentiate the teams, four different “team production functions” are assu m ed . I follow Kesenne (2007) and assu m e that the form ation of the team s consists of a certain nu m ber of S- and N -p layers. The S- and N -p layers are consid ered as being from alm ost com p lem ents, to alm ost su bstitu tes. All S-p layers are equally “Super” and all N -players are equally “Normal”3
. Team s u se their S- and N - p layers in ord er to “p rod u ce” p oints. The su p p ly of S- and N -p layers is u nlim ited . Since the valu e of the m arginal p rod u ct of p layers can’t be observed or m easu red , team s have a certain w age stru ctu re, from very d isp ersed t o very com p ressed . The w ages is the p olicy variable of the m od el. It is assu m ed that team s have no other fixed costs (like m anagers or other facilities); their only variable costs consist of the w ages they p ay to their p layers. It is also assu m ed that all team s receive sim ilar revenu es, either d irectly from UEFA, and / or from their p u blic, TV-rights and sp onsors and the w ages they p ay can’t be higher than their revenu es. All team s p lay a “Cou rnot” typ e gam e and m axim ize sim u ltaneou sly their p oints.
The m od el is rather general and can exp lain, not only the ow n p erform ance, bu t the effect to the other team s as w ell, even if they keep their w ages u nchanged . It can also
2
The third team continu es in the UEFA Eu rop a Leagu e.
3
Obviou sly, there are p layers w ho, objectively, belong to one category or the other. Perhap s, m ost p layers are neither that excellent to be classified in the first category, n or that “N orm al” to be
classified in the second category. In the football w orld , it is very com m on that the su p p orters of a team tend to overvalue their ow n players and und ervalue the competitor’s players. This is an emp irical issu e, left to su p p orters, to m anagers, to jou rnalists or even to those w ho d o research in efficiency analysis.
4
show w hether the tou rnam ent rem ains balanced and w hether team s w ho have m ore S-p layers can collect m ore p oints.
Let P1, P2, P3, P4, be the p oints collected by the team s; let S1, S2, S3, S4, be the nu m ber of S-p layers, and N1, N2, N3, N4 the nu m ber of N -p layers the team s u se in these m atches. Let w1, w2, w3, w4, be the victories of the team s, d1, d2, d3, d4, the d raw m atches and l1, l2, l3, l4, the losses (d efeats) of the team s. All these 24 variables are p ositive integers.
Since all firm s aim at m axim izing p oints (sim u ltaneou sly), their objective fu nction is given by (1), u nd er the follow ing constraints.
The key constraints that differentiate teams are the teams’ “production” functions , (2a – 2d ). All fu nctions are of Constant Elasticity of Su bstitu tio n (CES) typ e of d egree one, w ith d ifferent elasticity. The fu nction of team 1 is closed to Leontief, i.e. very low elasticity of su bstitu tion betw een its S- and N -p layers4. Team 4 on the other hand , has a closed to Cobb-Dou glas “p rod u ction” fu nction, i.e. alm ost excellent elasticity of su bstitu tion, team 2 is close to team 1, w hile team 3 is close to team 4. The u se of S- and N -p layers for each team w ill be end ogenou sly d eterm ined from the op tim al solu tion. When the team form ation has been d eterm ined , it is assu m ed that the sam e team w ill be u sed for all six m atches, u nless the w age stru ctu re ha s changed . Team s of cou rse change the com p osition of their p layers for tactical reasons, su ch as if they p lay aw ay against a stronger team or if they p lay at hom e against a w eaker team , or becau se som e of their p layers m ight be inju red or p u nished and are not available for a p articu lar m atch. The m od el neglects su ch p ossibilities. The m od el also assu m es that p layers w ho are not u sed (becau se the roster of team s consists of m ore than 11 p layers), receive zero w ages. N otice that, d u e to the integer constraint of p layers, the com p osition of the team can rem ain u nchanged , even if the w ages change. ti is the efficiency p aram eter and ai is the d istribu tion p aram eter .
Constraints (3a – 3d ) restrict the nu m ber of team p layers to 11. In reality, even for top and w ealthy team s, the nu m ber of S- is often low er than the nu m ber of N -p layers, w hich is given by constraints (4a – 4d ). Desp ite the fact, that su ch cond ition is not necessary, it is stated exp licitly in ord er to sp eed u p the solu tion of this com p lex m od el.
As is w ell know n, victories are w orth 3 p oints; d raw s are w orth 1 p oint and losses, zero p oints. Thu s, the nu m ber of p oints collected to every team is given by constraints (5a – 5d ).
4
In reality, an S-p layer can su bstitu te an N -p layer m ore easily than the reverse. I assu m e that the elasticity of su bstitu tion is u nch anged , irresp ectively if the S- su bstitu tes the N -p layer, or if the N - su bstitu tes the S-p layer.
5
parameters wage w w t j i j i Binary d d w w Integer l d w N S P d a N w S w P P P d a P d a l d w d a w w w d a d d d l a pairwise d w w l a pairwise d w w d a d w P d a N S d a N S N S t P d a N S t P N S t P N S t P t s P i i i i N S i i a i j h j i a i j h j i i i i i i i i N i S i i i i i i i i i i a j i h j i i a j i h j i a i j a i j h j i h j i a i j h j i i i i i i i i i i
, 1 0 0 , 4 ,.., 1 , ) , , , ( ) , , , , ( ) 13 13 ( , 0 5 . 0 12 ) 12 ( , 36 ) 11 11 ( , 18 ) 10 10 ( , 6 ) 9 9 ( , ) 8 8 ( , ) 7 7 ( , , 1 ) 6 6 ( , , 1 ) 5 5 ( 3 ) 4 4 ( ) 3 3 ( 11 ) 1 ( ) 2 2 ( ) 1 ( ) 1 ( ) 1 ( . . ) 1 ( max , , , , 2 4 1 , , , , , , , , , , 1 . 0 1 1 . 0 4 4 1 . 0 4 4 4 4 5 . 0 1 5 . 0 3 3 5 . 0 3 3 3 3 10 1 10 2 2 10 2 2 2 2 100 1 100 1 1 100 1 1 1 1 4 1 Each one of the 12 m atches in the grou p of fou r team s, end s either w ith a hom e team victory,wih,j,w ith an aw ay team victory,waj,i,or w ith a hom e team d raw ,dih,j,w hich is obviou sly equ al to the aw ay team d raw ,daj,i. Bu t, in ord er to id entify the correct p air of team s w hich p lay d raw at hom e and / or d raw aw ay , w e m u st sep arate the hom e team d raw from the aw ay team d raw , i.e. w e need 24 ad d itional constraints (6a – 7l). The first 12 constraints (6a – 6l) relate each p air of team s to hom e team d r aw and the rem aining 12, (7a – 7l), relate to the aw ay team d raw . If for instance d1h,2 1,(and consequ entlyw1h,2 w2a,1 0), from constraints (6a) and (7a) w e are ensu red that
, 1 1 , 2 a
d as w ell. If that m atch end s w ith a hom e or aw ay victory, it im p lies that . 0 1 , 2 2 , 1 a h d
6
m atch resu lts, su ch as d1h,2 d2a,10.5 and w1h,2 w2a,1 0.25. Therefore we require that all p ossible m atch resu lts are binary as w ell.
Obviou sly, the d raw s (and the victories) for each team is the su m of all p ossible d raw s (and victories) against all other team s. Constraints (8a – 8d ) ensu re that the nu m ber of d raw s in constraints is alw ays an integer (and actu ally even) nu m ber, or zero. For instance, w hen team 1 p lays only one m atch d raw , it is d11.If the d raw m atch w as a hom e m atch against team 2, it m u st be d1h,2 1,(from 8a), and 1,
1 , 2
a d
(from 8b), as w ell. In that case, if team 2 d oes not p lay another m atch d raw , it m u st be d2 1as w ell. Of cou rse, if team 2 p lays three d raw m atches, i.e. d2 3,it im p lies that team 2 m u st have p layed d raw against the other team s as w ell. A sim ilar interp retation ap p lies for the victory constraints (9a – 9d ).
Since each team p lay six m atches, there are 6 p ossible resu lts from its gam es, i.e. constraints (10a – 10d ) are requ ired . Constraints (11a – 11d ) show that no team can collect m ore than 18 p oints. In ad d ition to that, constraint (12) show s that the m axim u m nu m ber of p oints from all m atches is 36, com p osed w ith 12 victories (for tw o team s) and 12 d efeats for the other tw o team s.
Finally, constraints (13a – 13d ) ensu re non-negative profits. The revenu e fu nction is qu ad ratic in the p oints collected . All team s p ay the sam e, higher w ages to their S-p layers, and the sam e, low er w ages to their N -S-p layers. The initial S-p aram eters are:
8 i S w and 4.8, i N
w i.e. the N -p layers receive only 60% of the S-p layers’ w ages. Keep ing 8
i
S
w , the m od el is rep eated and solved for higher and low er valu es of the p aram eter .
i
N
w Obviou sly the w age d isp ersion or com p ression d oes not influ ence the p erform ance of the ow n team , bu t the p erform ances of the other team s too. The non-negative p rofits constraints are satisfied if each team collects at least 6 p oints. With less than 6 p oints there is no integer valu e of N - and S-p layers to ensu re non -negative p rofits.
The m od el is now com p lete and w as solved in Lingo, u sing Global Solver.
2.2 The Solution
I solved the m od el 124 tim es, i.e. 31 tim es per team , u sing 31 d ifferent ,
i
N
w starting from as low as 3.4 u p to 6.4, increased at a range of 0.1. When one team changed its
,
i
N
w all other team s keep their ow n
i N w u nchanged . When 6.4, i N w there is no feasible (integer) solu tion, becau se the non -negative p rofits constraint(s) are violated .
7
Table 1 d ep icts the ow n effects5 on p erform ance and team com p osition, w here the p olicy variable is
i
N
w . Som e p oints from the Table 1 are w orth m entioned .
The initial solu tion, w ith 8
i
S
w and 4.8,
i
N
w show s that the tou rnam ent is com p letely balanced , becau se each team w ins its three hom e m atches and collects 9 points. Despite the differences in their “production” functions, all four teams have a sim ilar team com p osition, consisting of 1 S-p layer and 10 N -p layers and each team m akes a p rofit of 11.5.
Table 1: Points m axim ization, su bject to non -negative p rofits i N w S1 N1 P1 S2 N2 P2 2 S3 N3 P3 3 S4 N4 P4 4 3.4 1 10 6 0.55 1 10 6 0.43 3 8 6 0.5 1 10 9 0.78 3.5 1 10 6 0.5 1 10 9 0.5 1 10 9 0.89 1 10 9 0.74 3.6 1 10 6 0.34 1 10 6 0.5 1 10 9 0.65 3 8 9 0.01 3.7 1 10 9 0.5 1 10 6 0.01 3 8 6 0.65 1 10 9 0.28 3.8 1 10 9 0.66 1 10 6 0.29 2 9 6 0.5 2 9 6 0.5 3.9 1 10 9 0.66 2 9 6 0.48 1 10 9 0.48 2 9 6 0.65 4.0 1 10 9 0.5 2 9 6 0.01 3 8 9 0.65 1 10 6 0.01 4.1 1 10 6 0.37 4 7 9 0.44 5 6 9 0.5 1 10 9 0.45 4.2 1 10 9 0.37 1 10 6 0.30 1 10 6 0.03 1 10 6 0.4 4.3 1 10 9 0.5 1 10 6 0.66 1 10 6 0.72 1 10 6 0.5 4.4 1 10 9 0.5 1 10 9 0.37 1 10 6 0.01 1 10 6 0.61 4.5 1 10 9 0.02 1 10 6 0.44 5 6 9 0.65 5 6 9 0.5 4.6 1 10 9 0.5 1 10 6 0.44 1 10 6 0.5 1 10 6 0.5 4.7 1 10 9 0.66 1 10 9 0.5 4 7 9 0.5 4 7 9 0.65 4.8 1 10 9 0.34 1 10 9 0.36 1 10 9 0.57 1 10 9 0.67 4.9 1 10 9 0.66 4 7 9 0.5 1 10 9 0.89 1 10 9 0.74 5.0 1 10 9 0.5 1 10 9 0.44 1 10 9 0.5 4 7 9 0.65 5.1 1 10 9 0.34 1 10 9 0.01 1 10 9 0.99 3 8 9 0.8 5.2 1 10 9 0.66 1 10 9 0.34 2 9 9 0.02 1 10 9 0.03 5.3 1 10 9 0.5 1 10 9 0.37 1 10 9 0.01 1 10 9 0.62 5.4 1 10 9 0.5 1 10 9 0.5 2 9 9 0.5 1 10 9 0.01 5.5 1 10 9 0.5 1 10 9 0.65 1 10 9 0.43 1 10 9 0.5 5.6 1 10 9 0.5 2 9 9 0.76 1 10 9 0.65 1 10 9 0.78 5.7 1 10 9 0.66 1 10 9 0.5 2 9 9 0.5 1 10 9 0.03 5.8 1 10 9 0.5 1 10 9 0.65 1 10 9 0.5 1 10 9 0.68 5.9 1 10 9 0.5 1 10 9 0.65 1 10 9 0.63 1 10 9 0.02 6.0 1 10 12 0.5 2 9 10 0.65 1 10 10 0.62 1 10 12 0.5 6.1 1 10 10 0.21 2 9 12 0.44 1 10 10 0.99 1 10 12 0.99 6.2 1 10 10 0.55 1 10 10 0.5 2 9 12 0.71 1 10 10 0.72 6.3 1 10 12 0.44 1 10 12 0.5 1 10 12 0.65 1 10 12 0.21 6.4 1 10 12 0.11 1 10 12 0.2 1 10 12 0.5 1 10 12 0.5 5
To save sp ace, I have exclu d ed the cross effects, i.e. effects on o ther team s. The fou r ad d itional Tables can be p rovid ed by the au thor at a requ est.
8
As exp ected , team 1 (closed to Leontief), never changes its team com p osition and u ses 1 S- and 10 N -p layers. On the other hand , it is team 3 that changes its team com p osition m ore frequ ent (in 11 ou t of 31 tim es) and not team 4, (closed to Cobb-Dou glas).
The m ost striking resu lt is that m ore S-p layers d o not alw ays increase the p erform ance of team s! This is d u e to tw o reasons. First, other team s m ight also p lay w ith m ore S-p layers too. Second the efficiency6 of the team s changes too. For instance, w hen wN4 4.7,both team 4 and team 2 u se 7 N -p layers and all team s collect 9 p oints and their resp ective efficiencies are alm ost sim ilar, t443.6, t2 38.6.
When wN4 6.0, both team s u se 10 N -p layers, team 4 collects 12 p oints, w hile team 2 collects 7 p oints, becau se their new efficiencies are t4 35.5, t2 7.5. Obviou sly, w hen the p erform ance d oes not increase, the p rofits of the team s are red u ced w ith m ore S-p layers. And the p oints collected p er team are not related to their d istribu tion p aram eter of p layers’ ai.
Finally, the tou rnam ent ap p ears to be frequ ently balanced . For instance, it is com p leted balanced in 23/ 31 cases w hen team 1 selects w ages; it is also com p letely balanced in 16/ 31 cases w hen team 2 selects w ages, and in 19/ 31 cases w hen team 3 or team 4 select w ages. And w hile in m ost cases the total nu m ber of p oints selected equ als to 36, there are som e d raw m atches w ith a total nu m ber of points equ al to 35 (alw ays w hen 6.0
i
N
w ).
2.2.1 The regressions from the optimal solutions
(i) The ow n effects
In ord er to find ou t how the w age d isp ersion influ ences the p erform ance of the team s, I ru n the follow ing tw o regression equ ations, based on all 124 op tim al solu tions. 2 2 1 1 ) 8 ( ) 8 ( ) 8 ( i i i N N i N i w w P w P .
The right hand sid e variable is the w age d ifference betw een the resp ective fixed
8 i S w and i N
w , w hile the d ep end ant variable is p oints collected . N egative (p ositive) -estim ates im p ly that the p erform ance of team s im p roves if the w age equ ality (inequ ality) betw een the S- and N -p layers increases. Sim ilarly, negative -estim ates
6
9
and p ositive -estim ates im p ly a U-typ e su ccess, i.e. either highly u nequ al or highly equ al w ages w ill im p rove p erform ance.
Table 2a su m m arizes the regression estim ates for every team . Based on the average nu m ber of p oints, team 1 and 4 are qu alified . It is clear that in the linear m od el, all team s im p rove their p erform ances w hen the w age d isp ersion d ecreases (all -estim ates are strongly negative). Consequ ently, the find ings su p p ort the Milgrom (1988) and Lazear (1989) hyp othesis.
Table 2a: OLS ow n estim ates Team 1 47 . 1 97 . 8 P Team 2 89 . 1 39 . 8 P Team 3 72 . 1 68 . 8 P Team 4 80 . 1 74 . 8 P 12.65** (19.24) 13.6** (18.52) 12.75** (15.64) 12.77** (14.16) -1.19** (-5.84) -1.68** (-7.39) -1.32** (-5.20) -1.30** (-4.65) 2 R 0.52 0.64 0.46 0.41 12.97** (5.44) 17.25** (6.73) 18.36** (6.70) 21.7** (7.88) -1.41 (-0.87) -4.25* (-2.43) -5.26** (-2.82) -7.58** (-4.04) 0.036 (0.14) 0.414 (1.48) 0.637* (2.13) 1.01** (3.38) 2 R 0.51 0.66 0.52 0.56
** significant at 0.01 level; * significant at 0.05 level; t -statistics is in p arentheses
Regard ing the qu ad ratic fu nction, the U-typ e su ccess for team s 3 and 4 is not rejected , over the relevant range (3.4,...,6.4).
i
N
w For instance, team 3 w ill m inim ize its p erform ance and collect abou t 7.5 p oints if it p ays 3.9,
3
N
w (i.e. if its N -p layers receive slightly less than 50% of the S-p layers w ages), it w ill collect abou t 8.5 p oints if it p ays the low est w age 3.4
3
N
w and it w ill collect abou t 11.5 p oints if it p ays the u p p er-lim it w age 6.4.
3
N
w Sim ilarly, team 4 w ill m inim ize its selected p oints (7.5), if it p ays 4.25
4
N
w .
(ii) The cross effects
In ord er to find ou t the p erform ance effects to the other team s, I ru n the follow ing tw o regression equ ations:
j i w w P w P j j j N N i N i , ) 8 ( ) 8 ( ) 8 ( 2 2 1 1
10
The estim ates are given in Table 2b. In the linear m od el, the estim ates are com p letely consistent w ith those in Table 2a. A d ecrease of w age d isp ersion in team i red u ces the p erform ance of team j (all -estim ates are strongly p ositive).
Table 2b: OLS cross estim ates
Linear Quadratic 2 R R2 Effect on Team 2 Team 1 7.19** (12.2) 0.64** (3.48) 0.27 9.08** (4.3) -0.69 (-0.5) 0.22 (1.14) 0.27 Effect on Team 3 8.20** (27.1) 0.22* (2.32) 0.13 5.87** (5.89) 1.85** (2.74) -0.26* (-2.43) 0.26 Effect on Team 4 6.99** (14.8) 0.59** (4.1) 0.34 4.57** (2.78) 2.3* (2.05) -0.27 (-1.53) 0.37 Effect on Team 1 Team 2 6.95** (13.2) 0.66** (4.05) 0.34 7.5** (3.93) 0.28 (0.2) 0.06 (0.3) 0.32 Effect on Team 3 6.87** (11.9) 0.66** (3.73) 0.30 2.61 (1.37) 3.66** (2.81) -0.48* (-2.32) 0.39 Effect on Team 4 7.60** (9.95) 0.62* (2.60) 0.16 5.14 (1.88) 2.35 (1.27) -0.28 (-0.94) 0.16 Effect on Team 1 Team 3 6.83** (9.21) 0.85** (3.69) 0.30 5.88* (2.20) 1.51 (0.83) -0.11 (-0.37) 0.27 Effect on Team 2 8.15** (27.3) 0.23* (2.53) 0.15 5.24** (5.72) 2.28** (3.66) -0.33** (-3.3) 0.37 Effect on Team 4 7.3** (14.3) 0.49** (3.13) 0.23 3.0 (1.82) 3.52** (3.13) -0.49* (-2.7) 0.37 Effect on Team 1 Team 4 7.91** (17.5) 0.34* (2.45) 0.14 6.59** (4.08) 1.27 (1.16) -0.15 (-0.8) 0.14 Effect on Team 2 7.57** (11.0) 0.49* (2.30) 0.13 1.37 (0.63) 4.86** (3.28) -0.7** (-2.97) 0.31 Effect on Team 3 6.78** (10.3) 0.73** (3.56) 0.28 2.84 (1.26) 3.5* (2.28) -0.45 (-1.82) 0.33
In the qu ad ratic m od el, som e estim ates are consistent w ith those in Table 2a, becau se a reversed U-typ e su ccess ap p ears. For instance, w hen team 3 p ays 6.4,
3
N
w w hile
team 2 keep s its initial w ages at 4.8,
2
N
11
w ou ld collect 11.5 p oints. The p air of team s (first entry d enotes w age d isp ersion of team i and second entry d enotes the p erform ance of team j) w ith a reversed U-typ e p erform ance are: (1, 3), (2, 3), (3, 2), (3, 4) and (4, 3), w hile in the rem aining seven p airs, there is no effect. N otice that team 3, w hich im p roves its ow n p erform ance by low or high w age d isp ersion (Table 2a), is affected negatively by all other team s too. To su m m arize the estim ates, w e conclu d e that team 1 (w ith the alm ost Leontief typ e), w hich collects the highest average nu m ber of p oints (8.97), im p roves its ow n p erform ance linearly w ith low er w age d isp ersion. And d esp ite the fact that the U-form ed w age d isp ersion d oes not im p rov e its ow n p erU-form ance, it is the only team that is not influ enced from low or high w age d isp ersions of other team s.
3. Empirical study
Obviou sly, to test the sam e hyp otheses w ith real d ata is alm ost im p o ssible, becau se there are many problems with the observed players’ wages and of course the team form ation. It w ill requ ire a hu ge am ou nt of tim e to refine all available d ata set by observing each ind ivid u al p layer , to exam ine if he w as inju red or p u nished and u navailable for som e m atches, or if he p layed only a sm all p art in a m atch. And even if su ch inform ation w ere relatively easy to collect, the w age d isp ersion of the team w ou ld vary d ep end ing u p on the various w eights one u ses for each p layer p articip ation in the m atches. To ad hoc exclu d e som e p layers (often the you ngsters w ith low w ages) m ight be erroneou s too, becau se som etim es m anagers d o not p lay the “expensive” players and deliberately use “cheap” players in some matches. Tw o very sm all d ata sets have been collected , assu m ing sim p ly that all p layers, for w hom w age observations exist, are available to p lay, and all p layers’ w ages have th e sam e w eights, irresp ectively if they p layed or not. The team sp irit or the envy of p layers exists for the entir e team roster and not necessarily for those p layers w ho are field ed . The first d ata set consists of 32 Eu rop ean team s (496 p layers) from 48 UCL grou p m atches, p layed in 2011. The aggregate statistics are show n in Ap p end ix (Table A). As is show n, for som e team s there are accu rate observations for at m ost 10-13 p layers. The second d ata set consists of 40 Italian team s from the Serie A (abou t 1015 p layers), over tw o seasons 2010-11/ 2011-12, obtained from the follow ing sites: http :/ / w w w .football-m arketing.com / 2010/ 09/ 07/ italian -serie-a-w age-list/ ,
http :/ / w w w .xtratim e.org/ foru m / show thread .p hp ?t=261972. N otice that 34 of these Italian team s ap p ear in both seasons, w hile the rem aining 6 team s ap p ear only in one season (3 of them relegated in the first season and 3 of them ad vanced in the second season). Since the team roster of these 34 team s change and / or som e w age contracts are re-negotiated , the w age d isp ersion w ithin the sam e team varies from year to year.
12
Table 3 su m m arizes the OLS estim ates for these d ata sets. In UCL, the w age d isp ersion is m easu red by Coefficient of Variation (CV) and by Mean Absolu te Deviation (MAD) as w ell; as control variables, tw o d ifferent p roxies are u sed , the team valu e (V) and the resp ective UEFA r anking (R); in Serie A, the w age d isp ersion is also m easu red by CV and by Coefficient of Disp ersion (CD) w hile as a control variable the average w age (W) p er team is u sed .
In case (i), the nu m ber of p oints d oes not au tom atically reflect qu alification or elim ination from the tou rnam ent, becau se it d ep end s on the grou p . For instance in grou p D, the elim inated third team ’s p oints (Ajax) are id entical to the second team ’s (Lyon), bu t Lyon had a better goal d ifference against Ajax. Sim ilarly, w hile Manchester City elim inated w ith 10 p oints, fou r other team s in other grou p s qu alified w ith less p oints. Consequ ently, in case (ii), a d u m m y variable w as introd u ced w ith the follow ing valu es: 1 for the tw o qu alified team s p er grou p ; 0.5 for the third elim inated team and 0 for the fou rth team .
Table 3: OLS estim ates for UCL and Serie A
UCL: (i) Point s
2 R R2 M A D V CV V 4.205** (4.50) 1.775 (0.78) 0.014 (1.89) 0.45 4.66** (2.61) -0.45 (-0.09) 0.02** (4.91) 0.44 M A D R CV R 3.066** (2.66) 3.056 (1.93) 0.044* (2.16) 0.47 2.14 (1.12) 2.865 (0.6) 0.07** (4.60) 0.41
UCL: (ii) (Dummy )*(Point s)
M A D V CV V 2.24 (1.79) 4.37 (1.45) 0.008 (0.75) 0.35 1.72 (0.74) 2.76 (0.42) 0.02** (3.61) 0.30 M A D R CV R 0.64 (0.43) 3.52 (1.73) 0.051 (1.95) 0.41 -1.13 (-0.47) 5.55 (0.92) 0.08** (4.17) 0.37
It alian Serie A: Point s
CD W CV W 42.73** (6.81) -0.08 (-0.68) 14.34** (5.99) 0.50 43.49** (6.49) -0.08 (-0.62) 14.44** (6.01) 0.50
** significant at 0.01 level; * significant at 0.05 level; t -statistics is in p arentheses.
Both d ata sets show sim ilar estim ates, w ith rather low exp lana tory p ow er. N one of the d isp ersion variables is statistically significant from zer o. On the other hand , as exp ected , stronger or richer team s p erform bet ter. Moreover, since both sam p les are very sm all and the p layers’ statistics are not refined , it is d ifficu lt to d raw certain conclu sions.
13
4. Conclusions
The p u rp ose of this p ap er w as to d evelop a kind of “general equ ilibriu m ” m od el to investigate if team s p erform better or w orse w hen they p ay rather com p ress ed or m ore d isp ersed w ages to their S- and N - qu ality p layers. In the m od el, fou r d ifferent football team s com p ete, in a tou rnam ent like the U CL grou p m atches, to m axim ize their p oints and qu alify in the next grou nd . Som e key featu res of the m od el are the non-linearity of “p rod u ction” fu nctions (w hich are of CES typ e w ith d ifferent elasticity of su bstitu tion am ong p layers) and a nu m ber of various integer variables (like p layers, p oints, victories and d raw s). Assu m ing a high w age level for the im p licitly solved S-qu ality p layers in every team and a rather large range of low er w ages for the im p licitly solved N -qu ality p layers, global optim al solu tions w ere obtained .
In m ost cases, the team form ation is com p osed by 1 S- and 10 N -p layers. Moreover, as it often hap p ens in football, team s w hich field m ore S- p layers d o not alw ays p erform better than team s w ith ju st 1 S-p layer! This is m ainly d u e to d ifferences in absolu te and / or relative efficiency the team s.
Desp ite the fact that all fou r team s p erform alm ost equ ally w ell and the w age p aram eters lead to a rather balanced tou rnam ent, team 1, w ith the low est elasticity of su bstitu tion betw een S- and N - p layers, collects on average, slightly m ore p oints than all others (follow ed by team 4, w ith the highest elasticity of su bstitu t ion).
Using the collected p oints from the resp ective op tim al solu tions, linear and qu ad ratic regressions w ere ru n to exam ine the w age d isp ersion on (i) the ow n effects and (ii) the cross effects. All fou r team s im p rove their ow n p erform ance if the w age d isp ersion d ecreases. In m ost cases, the d ecrease in w age d isp ersion d eteriorates the p erform ance of other team s as w ell. In ad d ition, som e evid ence on the U-typ e ow n su ccess and a reverse U-typ e cross su ccess ap p ears. Consequ ently, w hile highly d ep ressed w ages im p rove p erform ances linearly, interm ed iary w age d isp ersion is w orse than a highly d isp ersed one.
On the other hand , u sing tw o sm all d ata sets, from the UCL and the Italian Serie A, the observed w age d isp ersion from all p layers of the team s w as not statistically significant from zero, w hen other controlled variables (like the ranking or the valu e of team s) w ere inclu d ed in the regressions.
The m od el can be easily extend ed to catch other im p ortant asp ects. For instance, instead of m axim izing p oints, su bject to non -negative p rofits, team s can m axim ize p rofits. One can also increase the “p rice” p aram eter of p oints from 12 to 15, so that team s can collect m ore than 12 p oints that som e global solu tions p rovid ed . Another
14
interesting extension can be to allow d ifferent team s com p ositions in d ifferent matches, or combine the “production” functions per match, to find out if the “Leontief team” beats or is defeated by the “Cobb-Douglas team”, at home or away, and try to collect ap p rop riate m atch d ata to test the d erived hypothesis. Finally, since the selected CES fu nctions d o not let team s u se variou s tactical d isp ositions of p layers, it w ou ld be d esirable to stress the field p ositions of the S- and N -p layers. That can be sim ilar in sp irit to H irotsu & W right (2006), w ho ap p lied a N ash -Cou rnot gam e to figu re ou t the w in p robabilities of the 4-4-2 strategy over the 4-5-1 one.
15
Table A: MAD, CV, UEFA Ranking, Team valu e in m . of € and Points
Teams per group Observed players MAD (1) CV (2) Rank (3) Value (4) Points (5) FC Bayern 24 1.8196 0.5517 122.507 359.95 13 N ap oli 24 0.4203 0.6453 39.853 194.2 11 Man. City 16 1.3656 0.3051 61.507 467.0 10 Villarreal 16 0.2875 0.3647 78.551 142.6 0 Inter 26 1.3231 0.6479 102.853 246.85 10 CSKA 16 0.2625 0.2153 80.566 137.3 8 Trabzonsp or 13 0.2793 0.3130 20.115 87.8 7 Lille 13 0.3562 0.2580 38.802 119.75 6 Benfica 16 0.2672 0.2189 86.835 168.2 12 Basel 13 0.2414 0.2704 53.360 48.3 11 Man. United 16 0.9422 0.3329 141.507 415.0 9 Otelu i Galati 13 0.1402 0.2970 7.764 18.15 0
Real Mad rid 16 1.5078 0.4148 110.551 542.0 18
Lyon 15 0.9173 0.3394 94.802 152.2 8
Ajax 13 0.4911 0.3211 57.943 97.35 8
Dinam o Zagreb 13 0.2734 0.2972 24.774 44.8 0
Chelsea 16 1.3500 0.3127 122.507 381.0 11
Bayer Leverku ssen 13 0.5609 0.2907 59.403 137.0 10
Valencia 13 0.3030 0.1755 83.551 180.0 8
Genk 13 0.2367 0.2882 12.480 47.9 3
Arsenal 15 0.8089 0.3250 113.507 299.25 11
Marseille 13 0.5882 0.2520 84.802 140.65 10
Olym p iacos 15 0.3067 0.3118 61.420 81.4 9
Boru ssia Dortm u nd 16 0.6031 0.4752 30.403 179.75 4
Ap oel 13 0.2604 0.3221 33.599 14.95 9 Zenit 13 0.3172 0.2421 79.066 155.2 9 Porto 13 0.3976 0.2730 97.835 210.0 8 Shakhtar 13 0.4391 0.4539 83.894 143.45 5 Barcelona 16 1.6711 0.3680 151.551 579.0 16 AC Milan 25 1.2758 0.6483 88.853 266.3 9 Plzen 13 0.2166 0.3445 14.070 163.0 5 Bate Borisov 13 0.1112 0.2403 29.641 17.05 2
Notes: (1) & (2): For the three Italian teams, the annual wages are found in: http://www.xtratime.org/forum/showthread.php?t=261972 ; for all other teams, searching extensively various sport sites, forums and the teams’ official sites; most values for the following teams are uncertain and for the non- Euro teams, estimated with various exchanges rates into €: Trabzonspor, Otelui Galati, Dinamo Zagreb, Genk, Apoel, Plzen, Bate Borisov.
(3): http://kassiesa.home.xs4all.nl/bert/uefa/data/method4/trank2012.html (4): http://www.transfermarkt.co.uk/
16
References
Arrow , K., et al (1961): Cap ital-labor su bstitu tion and econom ic efficiency, Review of Economics & Statistics, 43, pp. 228-232.
Avru tin, B. M. & P. M. Som m ers (2007): Work incentives and salary d istribu tions in m ajor leagu e baseball, A tlantic Economic Journal, 35, pp. 509-510.
Breu nig, R., et al (2012): Wage d isp ersion and team p erform ance: a theoretical m od el and evid ence from baseball, Discussion Paper N o. 663, Centre for Economic Policy research, The A ustralian N ational University.
Fehr, E. & K. M. Schm id t (1999): A theory of fairness, com p etition, and coop eration, The Quarterly Journal of Economics, 114, pp. 817-868.
Franck, E. & S. N ü esch (2007): Wage d isp ersion and team p erform ance – An em p irical p anel analysis, W orking Paper N o. 73, Institute for Strategy and Business Economics, University of Z urich.
H irotsu , N . & M. B. Wright (2006): Mod eling tactical changes of form ation in association football as a zero-su m gam e, 2 (2), article 4, Journal of Quantitative A nalysis in Sports, The Berkeley Electronic Press.
Kesenne, S., (2007): Does a w in bonu s help to increase p rofit or w ins in p rofessional team sp orts? W orking Paper, Economics Department, University of A ntwerp.
Lazear, E. P. (1989): Pay equ ality and ind u strial p olitics, Journal of Political Economy, 97, pp. 561-580.
Lazear, E. P. & S. Rosen (1981): Rank-ord er tou rnam ents as op tim u m labor contracts, Journal of Political Economy, 89, pp. 841-964.
Levine, D. I. (1991): Cohesiveness, p rod u ctivity, and w age d ispersion, Journal of Economic Behavior and Organization, 15, pp. 237-255.
Milgrom , P. R. (1988): Em p loym ent contracts. Influ ence activities, and efficient organization d esigns, Journal of Political Economy, 96, pp. 42-60.
Pap ahristod ou lou , C. (2008): An analysis of UEFA Cham p ions Leagu e m atch statistics, International Journal of A pplied Sports Science, 20 (1) pp.67-93.
17
Pokorny, K. (2004): Pay – bu t d on’t p ay too m u ch: An exp erim ental stu d y on the im p act of incentives, W orking Paper, (available at http :/ / ssrn.com / abstract=614742)
Sim m ons, R. & D. J. Berri (2011): Mixing the p rinces and the p au p ers: Pay and p erform ance in the national basket association , Labour Economics, 18, pp. 381-388. Torgler, B., et al (2008): Looking aw kw ard w hen w inning and foolish w hen losing: Inequ ality aversion and p erform ance in the field , W orking Paper N o 2008-11, Center for Research in Economics, M anagement and the A rts, Basel.
Wisem an, F. & S. Chatterjee (2003): Team payroll and team p erform ance in m ajor leagu e baseball 1985-2002, Economics Bulletin, 1(2), pp. 1-10.
http://www.football-marketing.com/2010/09/07/italian-serie-a-wage-list/ http://www.xtratime.org/forum/showthread.php?t=261972 http://kassiesa.home.xs4all.nl/bert/uefa/data/method4/trank2012.html http://www.transfermarkt.co.uk/ http://www.uefa.com/uefachampionsleague/season=2012/matches/round=200026 3/index.html
