Int. Rev. Econ. (2007) 54:345–370 DOI 10.1007/s12232-007-0022-5
Competitive balance in football leagues when teams have different goals Nicola Giocoli
Published online: 18 September 2007 Ó Springer-Verlag 2007
Abstract In the standard two-team model of professional league sports it is shown that if teams have different objectives (the maximization of, respectively, wins and profits) the competitive balance conditions get worse with respect to the case when teams share the same goal. A similar, though less clear-cut, result obtains in the three-team setup. These outcomes call for policy measures to restore the balance. Three such measures are examined here: market-size-based revenue sharing, general salary cap and team-specific salary cap. It is shown that, contrary to the same-goalfor-all case, each of them may bring more intra-league competition. A ranking of the three measures is also suggested. Keywords Team sports Competitive balance Revenue sharing Salary cap Asymmetric paternalism JEL Classifications
D29 L21 L83
1 Introduction The paper presents a simple extension of the basic two- and three-team model of a professional sports league with the aim of capturing a peculiarity of European team sports in general, and of Italian football in particular, which has so far been neglected in the literature. The canonical model, as envisaged, for example by Fort and Quirk (1995) and Szymanski (2003), suggests two possible behavioral assumptions about professional teams: clubs are supposed to maximize either their profits (profit-maximizing, PM, behavior) or the number of seasonal wins N. Giocoli (&) Department of Economics, University of Pisa Faculty of Law, via Curtatone e Montanara 15, 56126 Pisa, Italy e-mail:
[email protected]
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(win-maximizing, WM, behavior). What matters most, behavior is assumed to be homogeneous, so that teams are either all PM or all WM: the former being predicated to mirror the reality of US professional leagues, the latter being viewed as better capturing the situation of European team sports. If we define the league’s competitive balance as the deviation from the ideal case where all teams win exactly the same proportion of matches during the season and if we accept the traditional assumption of sports economics literature that each team’s revenues are concave with respect to the team’s win percentage on account of the fans’ diminishing interest, and willingness to pay, for events whose outcome can be easily predicted (so-called uncertainty of outcome hypothesis), the well-known result is that a league’s competitive balance is higher under the all-PM case than under the all-WM one.1 This because rational, profit-oriented teams internalize the negative effect on their revenues of an ‘‘excessive’’ number of wins and thus never fully exploit any technical or athletic advantage that might stem from their financial strength vis-a`-vis their weaker opponents; on the contrary, rich WM teams simply try to extract as many wins as possible from such an advantage, eventually winning almost all matches and thus earning so little revenues that they barely manage to break even or, worse, incur in (possibly huge) economic losses that owners are called to stand at the end of each season. What is suggested in the paper is that a more realistic characterization of European professional leagues is that of ‘‘mixed’’ leagues, namely, of an environment where, while some teams still pursue the traditional win-maximizing goal, some others have modified their behavior in the direction of straightforward profit-maximization (so-called Americanization of European sports: see Hoehn and Szymanski 1999). In particular, it is argued that this characterization may capture the recent development in the main Italian football league: the Serie A: where, in the face of the dramatic financial imbalances caused to WM teams by the rising cost of the players’ talent, a few clubs seem to have shifted to a profit-oriented pattern of behavior. Hence, we propose a modified version of the standard model, where one team is PM, while the other(s) is (are) WM. The peculiarity of our simple comparative statics exercise is that the team that is assumed to embrace a profit goal is either the small (in the 2-team setup) or the medium (in the 3-team case) one in terms of revenue potential and market size (which in turn depends on, say, the team’s urban area of reference, the number of its fans, the latter’s ability and willingness to pay, the team’s skills in exploiting its merchandising possibilities, etc.). The rationale is straightforward: starting from an all-WM situation, where the equilibrium competitive balance entails a larger winpercentage for the big team because of its higher capacity to purchase technical– athletic talent, the small or medium team’s owner may reasonably ask herself what is the point of pursuing a sporting goal (say, winning the league’s title) she would never achieve, while at the same time forfeiting the possibility to earn some profits. 1
See, however, Fort and Quirk (2004) who deny any direct relation between teams’ objective and competitive balance, though by applying a slightly different model (for a discussion and delimitation of this result, see Kesenne (2004). Indeed, Fort and Quirk’s paper contains one of the best comparative studies of the two homogeneous cases; yet they still fail to address the more realistic ‘‘mixed’’ case. The present work aims at filling this gap.
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Hence, there is a strong incentive for the small or medium team to shift to a PM behavior, at the only price of a further—but possibly irrelevant from the end-ofseason outcome’s viewpoint—reduction in its win percentage. This, again, seems to capture a feature of the Italian Serie A (Baroncelli and Lago 2006). On one side, we have big teams’ owners who stick to a WM behavior, sometimes in the eventual pursuit of other, non-sporting goals (such as personal prestige and a positive spillover on their other business, or non-business, activities), and thus go on spending large amounts of money in the effort to secure wins, caring not too much about end-of-season financial losses which they are always willing to foot.2 On the other side, we have some medium and small teams’ owners who are well aware that they cannot really compete against the big clubs and who, above all, consider the management of a football club their main, and sometimes exclusive, business: these owners are led to adopt a PM behavior with the only constraint (not modeled here, but see Giocoli 2006) of not incurring in so many losses that their team risks being relegated in the league’s lower division. The outcome of our ‘‘mixed’’ model is a change in the league’s competitive balance with respect to both the ‘‘pure’’ (WM and PM) cases. In short, starting from an all-WM league and having the small or the medium team shifting to PM behavior should produce an increase in the win percentage of all the other teams on account of the reduction in the number of wins of the changing one. In the 2-team case this leads to an obvious worsening of the league’s CB, while in the 3-team setup, provided the behavioral shift is made by the medium club, we witness a softening of the competition for the top places and a hardening of the struggle to avoid the bottom positions. In the case of the Italian Serie A, these outcomes seem indeed to capture what emerges from casual observations (such as, say, the high frequency with which a few teams occupy the league’s top spots to the almost complete exclusion of the others) as well as from more systematic ones (such as the time pattern of the commonest measure of competitive balance, the standard deviation of seasonal wins).3 If this is so, what may be done to restore more balanced conditions on the playing field? The paper suggests three policy measures that may lead to this result: (1) market-size-based revenue sharing, (2) general salary cap, (3) team-specific salary cap. Remarkably, all are shown to be potentially effective in promoting conditions of more competitive balance in our ‘‘mixed’’ league. This goes against the traditional wisdom that, with the only exception of a general salary cap, no such measure may be capable of doing so (so-called invariance proposition, a cornerstone of team sports economics since Rottenberg 1956). Hence, the paper offers a normative, though idiosyncratic, justification for a change in the ‘‘rules of the game’’ of the top Italian football league. Moreover, it is suggested that, though all effective, the three measures may nonetheless be ranked in terms of either their 2
Note that this pattern of behavior is little affected by the club’s listing in the stock market as even a casual look at the financial statements of the very few Italian clubs who are actually listed immediately reveals: the only difference seems to be that in such cases the losses are spread through a larger number of shareholders.
3
The empirical analysis is carried on in a companion paper: Giocoli (2006).
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actual enforceability or their conformity to the main principle of asymmetric paternalism (Camerer et al. 2003). The content of the paper is as follows. In the next section, we review the basic ingredients of the standard team sports economics model. The so-called invariance proposition, the traditional benchmark for any policy evaluation is introduced in Sect. 3. The 2-team model in its two ‘‘pure’’ versions, namely, when teams are either both WM or both PM, is analyzed in Sect. 4. The fifth section presents the ‘‘mixed’’ model, where one of the teams—specifically, the small one—changes its behavior from WM to PM. In Sect. 6, we extend the analysis of the ‘‘mixed’’ case to the 3-team setup, this time assuming that the behavioral shift is made by the medium club. The seventh section offers our policy exercises in the 2-team case, as well as a ranking of the proposed measures. Section 8 concludes.
2 Team sports economics models: a review of the basic ingredients Most team sports economics (TSE) models share two general remarks and a specific assumption. The remarks concern the peculiar character of the productive process in that particular economic activity called ‘‘production of team sports events’’. First, this activity is characterized by the phenomenon of inverse joint production: in order to obtain a unit of output (here, a game or a tournament) two or more production processes, i.e., two or more firms (here, the sport clubs), are needed. It is still true that even in the case of team sports a firm is willing to outcompete its rivals because its revenues are increasing in the outcome of its production process (here, the results obtained on the game field, as each unit of output, viz., a game, is always assigned to one or the other of the firms who have contributed to its production).4 Yet, it is also true that the firms’ production functions exhibit a strong degree of complementarity, so much so that were a firm capable of conquering the whole market, ‘‘eliminating’’ all its rivals, it would see its revenues falling to zero on account of the impossibility to produce even a single unit of output on its own. Secondly, the outcome of a single firm’s production process in such a peculiar economic activity is summarized by an index, the performance on the field, which is subject to a tight aggregate constraint. Measuring each club i’s performance with the percentage wi [ [0,1] of games won with respect to games played in a given interval (say, a league’s season) the constraint is: n X i¼1
wi ¼
n 2
ð1Þ
(so-called adding up constraint), where n is the number of clubs in the given league. Thus, given the peculiar ‘‘rules of the game’’ of this particular economic activity, 4
In other words, while from the league’s viewpoint the output is the number of games played, from the club’s perspective, it is the number, or share, of games won. This assuming that each game has a winner and a loser. In case draws are admitted, they are considered as half win each.
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the overall output (the number of games, i.e. of wins)5 is predetermined and the only issue is how it will be divided among the firms (the league’s clubs). The assumption is about the consumers’ choice with respect to the outcome of this economic activity, that is, about the fans’ preferences for the games of a given team sport. First formulated by Rottenberg (1956) and Neale (1964), this assumption is known as uncertainty of outcome hypothesis (UOH) and claims that, generally speaking, the fans’ interest in any sport contest (not necessarily a team sports one) dwindles when its outcome is certain or, in any case, too easily predictable. In team sports, this amounts to saying that if wi ? 1, i.e. if it is almost sure that team i is going to win all its league games, consumers will reduce their demand for the ‘‘product’’, i.e. they will be less willing to spend their time and money to watch the games either live or in TV.6 From the club’s viewpoint, the UOH means that the revenue function is concave with respect to the win percentage: Ri ¼ Ri ðZi ; wi Þ
ð2Þ
where Zi is a vector of the other determinants of the club’s revenues, and oRi owi
2
[ 0; oowR2i \0: i
The assumption strengthens the cooperative element of the production process: not only it is impossible for a single club to produce the output ‘‘games won’’, but it is not even profitable to ‘‘produce’’ a wi too close to unity. It follows that the main notion in TSE literature is that of competitive balance (CB), namely, the degree of closeness in the win percentage of the different teams taking part in a given league. The highest CB is when wi ¼ 12 for each i, to indicate that each team wins exactly half of its games. In such a situation, we have the maximum uncertainty on a game’s outcome and thus, provided the UOH holds, the fans’ maximum willingness to dedicate their time and money to the event. The larger the deviation from such an ideal situation, the lower the degree of CB in the league, and thus the smaller the consumers’ demand and the clubs’ revenues. While basically sharing both the remarks and the UOH, TSE models may differ under several other respects. Let’s start from a hypothesis which is now standard in the literature, namely, that of considering the output ‘‘percentage of games won’’ as univocally determined, through a contest success function (CSF), by each club’s amount of technical and athletic talent. Indicating team i’s talent with xi and assuming that talent is measurable in homogeneous units, so much so that we can X P say that athlete a has more talent that athlete b if xa [ xb, we have xi ¼ xa ; a¼1 P xi as the total talent available where X is team i’s number of athletes. Taking X ¼ i
for all the clubs in a given league, a logit CSF is:
5
See the previous footnote.
6
For a more sophisticated analysis, dividing consumers into committed supporters and uncommitted TV watchers, with only the latter subject to the UOH, see Szymanski (2001).
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wi ¼
n xi n 2P xk
ð3Þ
k¼1
This is clearly a very strong, deterministic assumption. Talent is taken to be the only relevant input in the process leading to wi and each team’s share of wins is taken to directly increase with the talent at its disposal. It follows that each team is willing to add to its stock of talent by hiring it on the relevant market. Yet, such an incentive to buy is reduced by the UOH: the sign of the second derivative of the revenue function warns each club not to accumulate ‘‘too much’’ talent, lest its win percentage might become ‘‘excessive’’. The hypotheses behind (3) are quite common in the literature. From there, models can be classified along two dimensions,7 following specific assumptions on the total amount of talent X and the clubs’ goal. The first bifurcation is between models where the total talent supply is taken as fixed (fixed supply models) and models where the supply is variable. In the former the problem is once more purely distributional: any increment dxi of team i’s talent entails an equivalent (and, for simplicity, uniform) reduction of the available talent for all the other teams. In the latter (flexible supply models), such a perfect internalization of each dxi does not take place either because total talent supply is flexible or because, though fixed, it is nonetheless so large that no club is significantly affected by the variation of the talent available to another one. While the traditional TSE literature has so far focused just on fixed supply models,8 today ever more authors have reverted to flex supply models on account of their better fit with the reality of European football leagues.9 The second bifurcation pertains to the central issue of the next sections. What is a professional club’s goal? Is it the maximization of profit, as it is natural to assume when dealing with the US leagues? Or is it the maximization of wins, as it is suggested by the evidence of European sports? This is not the proper place to list the peculiarities of professional team sports in America,10 so I will just underline that they are fully consistent with the attribution to clubs of a profit maximizing behavior (so-called PM models). The European economists’ counterargument dates back to Sloane 1971: the ubiquitous presence in the Old Continent’s professional leagues of the mechanism of promotions and relegations11 and, more generally, the acknowledged relevance of sport as a social, non-strictly economic phenomenon in European societies make the assumption of profit maximization highly questionable and suggest the recourse to an alternative hypothesis. 7
At least, static models. It is curious to observe that the work pioneering most of the recent static literature, that of El-Hodiri and Quirk (1971), actually contained a dynamic analysis. For fully-fledged dynamic models see Palomino and Rigotti (2000), Szymanski and Valletti (2005).
8
See, e.g. Quirk and Fort (1992), Fort and Quirk (1995) and Vrooman (1995).
9
See Szymanski and Kesenne (2004) and Kesenne (2005).
10
See Hoehn and Szymanski (1999), Fort (2000) and Sanderson and Siegfried (2003). For a historical explanation of these peculiarities, see Cain and Haddock (2005).
11 Defined by the European Commission as a distinctive trait of European sports: see European Commission (1998).
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Some authors argue that this objection is today much less cogent, at least as long as European football is concerned, on account of the recent listing in the stock market of an ever increasing number of clubs. This fact alone is said to have forced the club’s managers to embrace a profit-oriented attitude, addressed at maximizing shareholders’ value and avoiding hostile takeovers. Hence, it is concluded, the USstyle PM models of clubs’ behavior are now perfectly suited to capture the working of European football leagues.12 Such a conclusion, however, is far from convincing, both because, as today, very few clubs are actually listed and because the recent experience suggests that precisely the listed ones have been the farthest from profitoriented management criteria.13 Bluntly, the fact that going public is just a way to finance a club well beyond its owner’s possibilities by exploiting the enthusiasm and naı¨vete´ of its most committed supporters is much more than a suspect. Thus, the most satisfying assumption as far as European team sports is concerned is still that clubs are after a sporting goal, namely, winning league titles or, more formally, maximizing the number, or share, of games won (so-called WM models). That this aim may be due to a not-fully-rational over-emphasis on immediate success14 or may in its turn be instrumental to the rational pursuit of other gains (say, in terms of popularity) to be spent by the club’s owner in her other, economic or non-economic, endeavors is an issue that needs not be tackled here, since in any case the club ends up being managed with the goal of winning as much as possible. It is immediate to remark that the two goals imply a different attitude towards the UOH. In the case of profit-oriented teams, that total revenues be concave with respect to the win percentage brings owners to take into account in their decision to purchase new playing talent the negative effect caused by dxi on the team’s marginal revenue. In a fixed supply setting this effect is worsened by the negative impact that any addition of talent for team i has on the available talent stock, and thus also on the other teams’ win percentage. Given the adding up constraint (1), this further increases team i’s wins and thus further decreases the marginal revenue created by the additional talent. In the case of win-oriented teams, instead, the relation between talent and marginal revenue is not so important: the club will try to win as often as possible and thus will strive to boost its talent stock, regardless of the consequences on the bottom line. This is the reason why it is customary to impose in these models a break-even constraint upon the team’s budget, or, alternatively, to fix the maximum amount of losses that the team’s owner may be willing to face (so-called deep-pocket owner assumption: see below, Sect. 5). Finally, the standard TSE models are closed by a set of assumptions on the clubs’ cost structure. The commonest hypothesis is that the only relevant cost that clubs have to bear in order to ‘‘produce’’ wins is that for purchasing the playing talent. 12
Hoehn and Szymanski (1999) and Fort (2000). But also see the skeptical remarks in Baroncelli and Lago (2006).
13 In the Italian case, a recent authoritative rating of Serie A clubs based on criteria of good management and financial performance assigns just one star, out of a maximum of five, to two of the three listed clubs, S.S. Lazio and A.S. Roma. See Il Sole–24 Ore, 24 June 2007, p. 22. 14 Modern behavioral economics has focused on this possibility: see Rabin (2002). For example, the owner might exhibit so-called present-biased preferences when comparing overspending now, in order to win next season’s title regardless of budgetary equilibrium, to a similar behavior in the future.
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This cost is supposed to be simply equal to cxi, where c is the constant average and marginal cost of talent. Such a cost structure is meant to capture the perfectly competitive nature of the talent market under the free agency rule now common to both American and, as an effect of the well-known Bosman ruling, European professional sports. In both environments, a talent free of contractual duties is available for being hired on the market with no additional cost in terms of compensation for its former club. Given that by assumption the unit of talent is uniform, so too must be its cost: the difference between a star and a mediocre athlete is in the higher number of units of talent embodied in the former, but no athlete has any market power on the supply of the single unit of talent.15 Similarly, the presence of several clubs competing to acquire the units of talent warrants that no relevant market power may exist even on the demand side.16 As to the other costs that are necessary to the life of a professional team sports club (general expenses, coach’s and staff’s salaries, playing facilities, etc.), it is customary in the literature to assume them away either as fixed or as a constant share of talent expenses. This is consistent with the hypotheses behind the CSF (3), namely, that playing talent is the only determinant of wins.
3 The invariance proposition In a standard setup with UOH, fixed talent supply (and fully internalized variations), profit-maximizing clubs, constant average and marginal talent cost, and, for simplicity, just two teams—so-called EH–Q–F model17—a basic proposition may be derived. It is the so-called invariance proposition, possibly the most well-known result in TSE. The proposition is the answer to a crucial policy problem. Under the UOH, it is necessary, if only to beef up the teams’ profits, to set the league’s rules such that a sufficient degree of CB be warranted despite the existence of, possibly very large, differences in the teams’ ability to hire talent. The easiest way to warrant a high CB is to establish that the big, i.e., rich, clubs (those with the largest number of supporters) subsidize the small, i.e. poor, ones. The subsidy may take different forms, from the sharing of gate- and TV right-revenues to the mechanism of draft (the rule establishing that teams hire new talent by following a reverse order of classification) to the, today no longer legal, reserve clause system (the constraints on the rights to the performances of those athletes whose contracts have just expired). Which of these or other possible forms of subsidy is the most effective in warranting a sufficient degree of CB? The surprising answer offered by the invariance proposition is that none of them may have any effect whatsoever on the CB of a 15 There are models where talent is differentiated between high-quality and standard-quality: see Kesenne (1999). The assumption is that only the former’s supply is fixed, while the latter’s is perfectly elastic. 16 More exactly, each player should be hired by the club where the value of her talent’s marginal product is highest and should be paid an amount equal to the second highest value of marginal product, the latter being the opportunity cost for any rational player. See Fort and Quirk (1995, p. 1272). 17
See El-Hodiri and Quirk (1971) and, more recently, Quirk and Fort (1992) and Fort and Quirk (1995).
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given league: the CB is indeed invariant with respect to the rules for allocating either the revenues or the talent because, with one single exception, no such rule is capable to modify the asymmetry among the teams caused by their different ability to pay for talent in the marketplace. The exception is given by the salary cap mechanism, that is, by the rule fixing the maximum amount that each team may spend for talent. Fort and Quirk (1995, p. 1278) have demonstrated that in a model with the above-mentioned features the salary cap is the only rule capable of correcting the asymmetry between the teams, thereby improving the league’s CB. The logic underneath the invariance proposition is as follows. Assume that each club is forced to leave part of its revenues to the other clubs: this causes an equivalent reduction in the marginal value of a talent increase. This means that a small club, viz., one with below-average talent, wins and revenues, will surely benefit in terms of revenues from the subsidy granted by the big clubs, viz., those with above-average talent, wins and revenues, but also that the additional revenue will be exactly compensated by the reduction in the marginal value of the talent that may be purchased with it. Any small team’s extra victory, in fact, is valued less in terms of both direct and indirect revenues, the former because a share will anyway have to be left to the other clubs, the latter because if big clubs win less often, their revenues will diminish and thus also the share transferred to the small ones. Exactly the opposite holds for the big club. Under the simplifying assumptions of the standard EH–Q–F model, the two effects fully compensate, so that the CB does not change.18 The proposition has obvious policy implications. First, far from favoring more balanced competitive conditions, the only effect of the subsidies paid by big to small teams, as well as of other similar measures, is to either distort or restrain intraleague competition. Hence, contrary to what has happened time and again in both the US and Europe, these measures should not deserve to be exempted from antitrust rules. Indeed, their true goal often turns out to be that of collusively boosting the clubs’ profits to the detriment of the players’ earnings. Second, though the only effective measure to favor the CB is the salary cap, it should also be underlined that such a measure is the only one suffering from a serious enforcement problem since no team has the incentive to obey it. Analytically, the difficulty is given by the fact that in a salary cap equilibrium each club has a different marginal revenue and so may wish to unilaterally deviate from the cap.19 To sum up, a laissez-faire attitude towards a league’s CB, combined with the strict enforcement of antitrust prohibitions, seems to be the policy advice stemming from the invariance proposition. The proposition has been subjected to intense empirical scrutiny. As summarized in Szymanski (2003), the outcome has been mixed, with a few confirmations and several refutations. The latter should cause no surprise if attention is paid to the strong, and rather questionable, assumptions underlying the EH–Q–F model where the proposition has been originally derived. Indeed, the efforts to prove the proposition in a setup different from that prevailing in American professional sports 18
For a formal proof of the proposition see the works quoted in the previous footnote.
19
For the details, see again Fort and Quirk (1995). Also see Kesenne (2000a) and below, Sect. 7.
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have failed: the result is not robust to any modification in the assumptions concerning the properties of either the revenue function or the talent supply or the clubs’ objective function, and not even to an increase in the number of teams. In short, the most recent literature has shown that the invariance result fails when there are meaningful revenue components which are not shared between the teams, or when the supply of talent is elastic, or when the teams’ goal is the maximization of wins, or, even more simply, when there are more than two teams. In all these cases, the existence of a subsidy in favor of the small clubs, or of any other measure aimed at regulating the market for talent, is effective in increasing the CB. Hence, today, the invariance proposition is considered little more than a useful benchmark. The goal of the next sections is to provide one further exception to the invariance proposition. What we assume is that the teams have different objectives. This simple modification of the standard model suffices to show that some policy measures may be effective in improving a league’s CB.
4 The model in the ‘‘pure’’ cases Let us assume that team i has the following total revenue function:20 Ri ¼ mi wi bi w2i ;
i ¼ 1; 2;
ð4Þ
with positive first derivative and negative second derivative with respect to win percentage wi. The function has two parameters mi and bi. The first measures the team’s market size, that is, its potential with respect to all possible revenue sources (gates, TV rights, merchandising). In a two-team league, we assume that m1 [ m2, i.e. that team 1 is bigger, in terms of revenue potential, than team 2.21 The second parameter bi captures the interest of team i’s fans with respect to competitive balance CB. According to the UOH, the greater the team’s win percentage, the lower its fans’ interest and thus the smaller the increase in revenues that the team may get from improving its win percentage. bi measures the relevance of this hypothesis by making it team-specific; it is a matter of life, in fact, that some teams enjoy very keen supporters (‘‘win or lose, it’s my side’’) while others rely on not-socommitted fans who give much more weight to playing quality and game uncertainty. Hence, bi is an inverse measure of the fans’ commitment: the higher its value, the stronger the relevance of the UOH. We know that the sum of the win percentages must satisfy the adding up constraint (1). With n = 2, the constraint is w1 + w2 = 1, or w1 = 1 – w2. Team i’s contest success function (CSF) has the logit form: wi ¼ 20
xi X
ð5Þ
See Kesenne (2000b, 2004).
21
Note that mi could be viewed as team i’s share of the total market of a given league sport. In such a case, we would have m1 + m2 = 1 and the analysis would be further simplified. I thank Raul Caruso for this suggestion.
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where, as before, xi measures the playing talent available for team i, with X = x1 + x2. The CSF means that team i’s win percentage depends on the share of total talent at its disposal. Differentiating the CSF with respect to x1, we get: dx2= dx1 x1 ow1 x2 ¼ ð6Þ ox1 X2 It is here that the assumption of the elasticity of total talent supply X comes into play. If the supply is fixed and every talent variation in a team is fully internalized by the other team (as it is obvious in case of n = 2), we can set dx2=dx1 ¼ 1; since any increase in team 1’s talent reduces of the same amount the talent available for x2 þx1 1 1 team 2, and vice versa. Substituting in Eq. (6), we get: ow ox1 ¼ X 2 ¼ X : Normalizing 1 total talent to 1 (i.e., X = 1)22 we obtain ow ox1 ¼ 1: Hence, we can simply set w1 = x1, and replace in every formula team 1’s win percentage with the amount of its talent. The same holds for team 2. Talent and wins become perfect substitutes, so that by choosing its own level of talent each team is actually establishing its own win percentage. Substituting in the revenue function, we get:
Ri ¼ mi xi bi x2i
ð7Þ
while average and marginal revenue become, respectively, Ri ¼ m i bi x i xi
ð8Þ
oRi ¼ mi 2bi xi oxi
ð9Þ
ARi ¼ MRi ¼
In case we give up the assumption of fixed supply, and take instead talent to be perfectly elastic, we have dx2=dx1 ¼ 0 : team 1’s talent increase has no effect on x
j i team 2’s total talent, and vice versa. It follows that (6) becomes ow oxi ¼ X 2 ; i; j ¼ 1; 2; owi xi xj i and marginal revenue is MRi ¼ oR oxi ¼ ðmi 2bi wi Þ oxi ¼ mi 2bi X X 2 ; i.e. nonlinear and, above all, always lower than in the fixed supply case. Given that the reduction of the marginal revenue 2 is smaller for the small team (this because the multiplicative factor the xj X is larger for MR2 than for MR1, since m1 [ m2 ) x1 X 2 [ x2 X 2 ) and given that, as we show below, the equilibrium win percentage is determined at the intersection of the marginal revenue curves, we may conclude that, generally speaking, an elastic talent supply, typical of European leagues, helps increase the chances of the small team, thereby improving the CB.
22 Note that this normalization is different from the adding up constraint w1 + w2 = 1. While the latter holds in any TSE model, the former stems from having taken as fixed the total talent supply and as fully internalized any change in its distribution.
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On the cost side, we simply assume that team i’s total cost is: Ci ¼ cxi;
i ¼ 1; 2
ð10Þ
namely, that either the team has no other costs apart from the players’ salaries or that these other costs are strictly correlated to the wage fund, so that the cost c of a unit of talent also includes a constant share of the other expenses. As noted before, the unit cost is assumed identical for all teams, to indicate the existence of a perfectly competitive market for talent. In case both teams aim at the maximization of wins (both-WM model), let’s assume that the purchase of talent is limited by a break-even constraint.23 It follows that team i’s problem is: Max wi s.t. Ri ¼ mi xi bi x2i ¼ cxi ¼ Ci xi
ð11Þ
that is, s.t. ARi ¼ mi bi xi ¼ c ¼ ACi;
i ¼ 1; 2
ð12Þ
Therefore, by equating the two average revenues, we obtain the following equilibrium for a league where all teams aim at maximizing their win percentage under a break-even constraint: ~x~1 ¼ m1 m2 þ b2 b1 þ b2
ð13Þ
~x~2 ¼ m2 m1 þ b1 b1 þ b2
ð14Þ
~c~ ¼ m1 b2 þ b1 m2 b1 b2 b1 þ b2
ð15Þ
The three equations indicate, respectively, the amount of talent (and thus the win percentage) of teams 1 and 2 and the average and marginal cost of talent in the bothWM case. Note that, in order to have a non-negative ~x~2 ; the restriction b1 > m 1 m 2
ð16Þ
must be satisfied (~x~1 is instead always positive due to the assumption m1 [ m2). The constraint (16) may be interpreted as requiring that the big team’s supporters must be sufficiently interested in the uncertainty of the competition. Indeed, such an interest must not be lower than the difference in the teams’ market sizes. In short, to have a meaningful solution, it is required that either the big team’s fans be not too much committed or the mi’s be not too different.
23 Nothing would change in case we set a maximum amount of money losses that the club may bear: see next section.
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In case both teams aim at the maximization of profit (both-PM model), the equilibrium is given by equating each i’s marginal revenue and marginal cost: MRi = mi – 2bixi = c = MCi. Solving the system, we get the following results: ^x^1 ¼ m1 m2 þ 2b2 2ð b1 þ b2 Þ
ð17Þ
^x^2 ¼ m2 m1 þ 2b1 2ð b1 þ b2 Þ
ð18Þ
^c^ ¼ m1 b2 þ b1 m2 2b1 b2 b1 þ b2
ð19Þ
The three equations indicate, respectively, the amount of talent (and thus the win percentage) of teams 1 and 2 and the average and marginal cost in the both-PM case. Note that, in order to have a non-negative ^x^2 ; the restriction 2b1 [ m1 – m2 must be imposed, but this is always true if the previous constraint (16) holds. Moreover, the non-negativity of ~c~ and ^c^ is warranted by: m1= [ 2 m2= ð20Þ b1 b2 By comparing the solutions of the two ‘‘pure’’ cases, it is immediate to verify that ~x~1 [ ^x^1 and ~c~ [ ^c^: Thus, in the both-WM case the CB worsens and the cost of talent increases with respect to the both-PM case. This is intuitive: in a league where all teams aim at being successful on the field, rather than at earning profits, the big teams may exploit their revenue advantage to purchase more talent and thus win more games without being put-off by the negative effect on their own revenues of the increase in the number of wins, as instead would happen in case of a purely economic goal.24 It is also obvious that in case all teams try to maximize their win percentage, the competition to acquire the available talent is tougher, and so the equilibrium level of talent cost is higher (Fig. 1). We can illustrate the solutions of the ‘‘pure’’ cases with the classic Fort and Quirk diagram (see, e.g. Fort and Quirk 1995, p. 1273). On the X-axis we measure the win percentage of team 1, while on the Y-axis we measure the teams’ average and marginal costs and revenues. The diagram shows the two equilibria, for both-WM and both-PM cases. It is apparent that in the former case the degree of CB worsens and the unit cost of talent increases with respect to the latter.
5 The ‘‘mixed’’ model The results of the previous section are quite standard. Here we develop a simple, though novel, comparative statics exercise in which, starting from a both-WM 24 As shown by Kesenne (2004), the counterintuitive result in Fort and Quirk (2004) (namely, the possibility of a better CB in the both-WM case) crucially depends on assuming a league where, for some unspecified reasons, it is the small team which actually dominates.
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AR1 MR1
AR2 MR2 AR1 AR2
MR1 WPM
c
c MR2
cˆˆ
X2 = 1
PM
xˆˆ1 x1
cˆˆ
X1 = 1
Fig. 1 Fort and Quirk diagram: equilibrium in the two ‘‘pure’’ cases
situation (which, as we argued before, seems to best capture the tradition of European leagues), one of the teams modifies its behavior by embracing profit, rather than on-the-field success, as its new goal. The model thus becomes a ‘‘mixed’’ one, with one club WM and the other PM. The peculiarity of the exercise is that we assume that the shifting club is the small one. This is for two reasons. First, there is the idea that the big team’s owner may be keener on sporting success because she has many more chances than her small club’s colleague to obtain different, not necessarily (or not directly) economic, kinds of return from becoming an icon of success and strength. Second, and most important, it is rational for the small team’s owner, who knows very well that the difference in market size, and thus in revenues, makes it very unlikely that her team may ever beat the big one, to ask herself why going on accumulating defeats while at the same time forfeiting some possible profits. It is apparent, in fact, that by embracing a PM behavior, the small team could at least obtain a positive economic return, though at the price of further reducing its win percentage.25 Therefore, let us assume that, starting from a both-WM case, team 1 (the big one) sticks to WM, while team 2 (the small one) shifts from WM to PM. Team 1 thus chooses x1 to satisfy AR1 = AC1, while team 2 sets x2 to satisfy MR2 = MC2. 25 The most recent data available on Italian Serie A clubs’ financial performances give some support to this assumption: of the six clubs closing 2005–2006 season with a non-negligible profit, four were relatively small ones (Udinese, Parma, Livorno and Ascoli: see Il Sole–24 Ore, 24 June 2007, p. 22). Note however that in the European environment, characterized by the promotion and relegation system, even a small club may rationally stick to WM behavior in order to avoid relegation. This is why in the next section we assume that in a 3-team model the shift to PM is made by the ‘‘medium’’ club, that is, one which is neither so small to risk relegation by reducing its win percentage nor so big to be a realistic title contender by sticking to WM behavior.
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AR1 MR1
AR2 MR2 AR1 AR2
MR1 WPM
MR2 E
cˆ
cˆ
PM
xˆˆ1 x1
X2 = 1
x1
X1 = 1
Fig. 2 Equilibrium in the ‘‘mixed’’ case
Solving the system and assuming a fixed (normalized to 1) talent supply, we obtain the following solution for the ‘‘mixed’’ case: x~1 ¼
m1 m2 þ 2b2 b1 þ 2b2
ð21Þ
x^2 ¼
m 2 m 1 þ b1 b1 þ 2b2
ð22Þ
^c~ ¼ 2m1 b2 þ b1 m2 2b1 b2 b1 þ 2b2
ð23Þ 26
It is immediate to verify that, if restriction (16) holds, it is always x~1 [ ~x~1 : Hence, the shift to PM of the small team worsens the CB with respect to both the ‘‘pure’’ cases. As to the average and marginal cost, which is always positive because of (20), we have ^c^\^c~\~c~; where the second inequality holds, again, if (16) holds. Thus, talent cost in the ‘‘mixed’’ case is, respectively, lower and higher than in the bothWM and both-PM ones. The following diagram shows the equilibrium in a 2-team, fixed-supply model when team 1 is WPM and team 2 is PM. As we can see, at E the CB is worse than in the two ‘‘pure’’ equilibria, while talent cost is intermediate (Fig. 2). What happens in case the big club fails to satisfy the break-even constraint? This is an all-too-known case in European sports in general, and in football in particular, where several owners have accepted to stand large and repeated losses in the (often vain) effort to achieve on-the-field glory. We can model the existence of such a 26
The tilde and the hat over c indicate the mixed nature of the solution.
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AR1 MR1
AR2 MR2
AR1dpo
Γ AR1 MR2
c
E
dpo
dpo
E
cˆ
x1
X2 = 1
c dpo
cˆ
x1dpo
X1 = 1
Fig. 3 Effect on the ‘‘mixed’’ case equilibrium of a deep-pocket owner
‘‘deep-pocket’’ owner by assuming that team 1 is not bound anymore to abide by a balanced budget because it can incur in a loss of up to C x1, with C [ 0, which is increasing in the win percentage (this is to capture the intuition that the more the wins, the larger the total loss the owner is willing to bear). The new constraint is therefore C1 = R1 + C x1, and the new equilibrium condition for the big WM team is: AR1 + C = m1 – b1x1 + C = c. The latter can be put together with the standard profit-maximizing condition for the small team, m2 – 2b2x2 = c. By solving the system we obtain the equilibrium win percentage m1 m2 þ2b2 þC ; and the equilibrium cost of talent cdpo ¼ for team 1, xdpo 1 ¼ b1 þ2b2 2m1 b2 þb1 m2 2b1 b2 þ2b2 C : It is easy to show that xdpo [ x~1 and cdpo [ ^c~ : thus, with 1
b1 þ2b2
respect to the ‘‘mixed’’-case benchmark, the existence of a ‘‘deep-pocket’’ owner boosts both her team’s win percentage, to the detriment of the league’s CB, and the cost oxdpo
1 [ 0; that is, team’s 1 equilibrium win of talent. Moreover, we also have: oC1 ¼ b1 þ2b 2 percentage goes up with the amount of money that the owner is willing to lose for every game won. The diagram below shows the effect of a deep-pocket owner on the league’s equilibrium. Recalling that the ‘‘mixed’’ case already exhibits a lower CB than the ‘‘pure’’ ones, it follows that the violation of the break-even constraint by one, or more, big win-maximizing clubs calls forth, here more than ever, the adoption of policy measures addressed to restore a sufficient degree of CB (Fig. 3).
6 The three-team ‘‘mixed’’ case In this section, we analyze the ‘‘mixed’’ variant of the EH–Q–F model in the case of n = 3. As noted before, a behavioral shift from WM to PM seems more plausible for
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a team which is not so small to sensibly worsen the risk of relegation as an effect of the reduction in its equilibrium win percentage: let’s call it the medium team.27 Hence, we assume that teams 1 and 3, the big and the small, stick to their WM behavior, while the medium team 2 shifts to PM. To simplify the analysis, we also assume that, while b1 ¼ b [ 1; it is b2 = b3 = 1. The intuition is that only the big club’s supporters are so ‘‘choosy’’ and so accustomed to victory to give a significant weight to the quality and uncertainty of competition, while the UOH is much less relevant for the small and medium clubs’ fans who are above all eager to see their side win. If all teams are WM, each with a revenue function like (7) and m1 [ m2 [ m3, the solution is: 2m1 m2 m3 þ 1 2b þ 1
ð24Þ
ðb þ 1Þm2 bm3 m1 þ b 2b þ 1
ð25Þ
ðb þ 1Þm3 bm2 m1 þ b 2b þ 1
ð26Þ
^
x1 ¼
^
x2 ¼
^
x3 ¼
^
^
It can be verified that x 1 [ 0 always, while the non-negativity condition for x2 is ^ absorbed by the more stringent one on x 3 : Hence, when the latter holds, i.e., when b[
m1 m3 ¼A 1 þ m3 m2
ð27Þ
(with 1 [ m2 m3 ) ðm2 m3 Þ 2 ð0; 1Þ)28 we have that all three clubs obtain a ^ ^ ^ non-negative share of wins. Moreover, it is easy to check that x1 [ x2 [ x3 : Let team 2 shift to PM behavior. The system’s solution becomes: 2ðm1 m3 Þ þ m1 m2 þ 2 3b þ 2
ð28Þ
bðm2 m3 Þ þ m2 m1 þ b 3b þ 2
ð29Þ
bðm3 m2 Þ þ 2ðm3 m1 Þ þ 2b 3b þ 2
ð30Þ
_
x1 ¼
_
x2 ¼
_
x3 ¼
27 Going back to the data in footnote 25, Udinese Calcio is a typical medium club (ninth in terms of revenues among Serie A 20 clubs) which ranked first for profits in 2005–2006. 28 In case this condition is violated, i.e., m2–m3 = 1, we get x3 ¼ 0 : the small club wins no games at all. But see above, footnote 21. ^
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_
If we check for non-negativity, we note that x1 [ 0 always, while, again, the non_ _ negativity condition on x3 is more stringent than that on x2 .29 Yet, the condition on _ ^ x3 is in turn absorbed by the even more stringent one on x 3 : In short, if (27) holds, all equilibrium win percentages in both the all-WM and the mixed models are nonnegative. The next step is to verify that, as we expect, the shifting club decreases its share _ ^ of wins to the advantage of the other two teams. It is immediate to verify that x2 \x 2 _ ^ _ ^ always, while x 1 [ x 1 and x3 [ x3 both hold when, again, condition (27) is satisfied. Moreover, the same condition warrants that the equilibrium level of talent cost is lower in the mixed case than in the all-WM one, on account of the weaker competition on the talent market when one of the clubs looks at the bottom line _ ^ ^ 2 þbm3 b more than at the season’s ranking: c \ c ; where c ¼ m1 þbm2bþ1 and _ 2m1 þbm2 þ2bm3 2b c¼ : 3bþ2 Finally, we check which club stands to gain more from club 2’s behavioral shift. _ ^ It turns out that the increase in the small club’s win percentage ðx3 x3 Þ is b times _ ^ that of the big club ðx 1 x 1 Þ: Given that b [ 1, it is the small team which gains more. The overall effect on the league’s CB is therefore ambiguous: on the one side, we have a reduction in the gap between the small club and all its rivals; on the other, the distance between the big and the medium team widens. Empirically, what we should observe is a diminished competition for the league’s top places (because by shifting to PM the medium teams can no longer effectively compete against the big ones) and a more intense competition to stay away from the bottom positions (i.e., to avoid relegation, as the medium teams’ win percentage may get dangerously close to that of the small teams).30 To summarize, in a simplified 3-team model \‘a la EH–F–Q, if the medium team shifts to PM, we have that the deviating team wins less games, the other two clubs win some extra games (the small club even more so than the big one) and the talent cost is lower than when all teams pursue the same WM goal. This under the condition that the preference intensity of the big club’s supporters for a balanced competition is strong enough to guarantee that the key parameter b satisfies (27).
7 Pro-CB policy measures in the ‘‘mixed’’ case In this section we propose another simple comparative statics exercise, by applying to the equilibrium of the 2-team ‘‘mixed’’ model three of the policy measures that
29 This under the further condition that m2–m3 \ 2 (which however is encompassed by the previous one that m2–m3 \ 1). In case the difference between the medium and the small clubs’ market sizes exceeds 2, ^ _ ^ _ the small club never wins: x3 ¼ x3 ¼ 0: Finally, if (m2 – m3 ) [(1,2 ), we have x3 ¼ 0 and x3 [ 0; i.e., the small club manages to seize a few games only after the medium club has loosened the competitive conditions by shifting to PM. 30
An empirical analysis along these lines for the case of the Italian Serie A is in Giocoli (2006).
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are most frequently suggested in order to improve a league’s CB.31 The measures are, in order, (1) a market-related form of revenue sharing, (2) a general salary cap, and (3) a team-specific salary cap. The first is a form of revenue sharing among teams which is based on the extent of each team’s market rather than, as is standard in the literature, on the sheer amount of its revenues. The idea is that teams with a larger-than-average market (in terms of fans, pay-TV subscribers, merchandising potential, etc.) should subsidize those with a less-than-average one, with the subsidy being a fixed percentage of the deviation from the average.32 The second is a rule requiring each team to spend for players’ salaries no more than a given share (to be determined through negotiations between the teams and the players’ union) of the league’s total revenues: this is a rather drastic measure that, regardless of their different market size, would place all teams on par with respect to their ability to purchase playing talent. The third is a rule in the same spirit of the second one, though with a softer approach. The idea is to establish each team’s cap as a share of its own revenues, so that the big team would still have the possibility to spend on talent more than the small one, but less so than absent the measure. In the literature based on the EH–Q–F model and its variants, only the second measure has been showed to be always effective in modifying a league’s CB. On the contrary, all three turn to be effective in our exercise. Hence, it may be concluded that the heterogeneity of clubs’ behavior provides another instance of violation of the invariance proposition.
7.1 Market-related revenue sharing Assume teams’ revenues are shared according to the following formula: Þ Ri ¼ Ri aðmi m
ð31Þ
is average market size. Thus, the where a [ 0 is the sharing coefficient and m sharing arrangement amounts to having clubs with larger-than-average market leaving part of their revenues to clubs with smaller-than-average market. This leaves each team’s marginal revenue unaffected, so much so that the profit-maximizing condition for PM teams is also unchanged. On the contrary, team i’s average revenue becomes non-linear, ARRS i ¼ m i bi x i
Þ aðmi m ; xi
and this affects the break-even condition of WM teams. In particular, since the sign RS oAR i average revenue depends on mi being larger or smaller than m; of oa i ¼ mm xi 31
The implicit assumption is that a team’s choice to be either PM or WM is given. Alternatively, we might envisage policies aimed at pushing teams towards either a PM or a WM behavior (whichever is considered more desirable from the league’s viewpoint). See on this Fort and Quirk (2004). 32 For example, the rule might require that revenues be redistributed from the clubs with a larger-thanaverage number of pay-TV subscribers to those with a smaller-than-average number.
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AR1 MR1
AR2 MR2
AR1
MR2 E
cˆ
cˆ
ERS
c RS
c RS
RS
AR1
x2=1
[a(m1–m2)/2bt]1/2
x1
RS
x1
x1=1
Fig. 4 Effect on the ‘‘mixed’’ case equilibrium of market-related revenue sharing
increases with a for small teams and decreases for big ones. Given our assumption that in ‘‘mixed’’ models it is big teams which stick to WM behavior, while small ones shift to PM, it follows that this form of revenue sharing is effective in improving a league’s CB and lowering the cost of talent. The figure shows the effect of a market-related revenue sharing rule. The ARRS 1 ffi curveqisffiffiffiffiffiffiffiffiffiffiffiffiffi bell-shaped for non-negative values of x1, with a maximum at x1 ¼ aðm1bm Þ: To prove that the equilibrium win percentage xRS 1 is lower than x~1 ; we first calculate: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm m 2b Þ ðm1 m2 þ 2b2 Þ2 2að2b2 þ b1 Þðm1 m2 Þ 2 1 2 RS ð32Þ x1 ¼ 2ð2b2 þ b1 Þ 2 þ2b2 Recalling by (21) that x~1 ¼ m1bm ; let m1 – m2 – 2b2 = H. The term under 1 þ2b2 the square root in (32) may then be written as H2 – B, where B = 2a (2b2 + b1) (m1 – m2)[ 0. Hence, the result of the square root, say, U, is lower than H : U \ H. Noting that m2 – m1 – 2b2 = – H, we may therefore re-write (32) as: HU xRS 1 ¼ 2ð2b2 þb1 Þ : The solutions are both positive, because the numerator is always
HU negative. Yet, the biggest of the two solutions, xRS 1 ¼ 2ð2b2 þb1 Þ ; is still lower than
x~1 because – H – U \ – 2H and so the whole fraction is lower than 2b2Hþb1 : This means that the league’s CB is improved by a market-related revenue sharing rule, q.e.d (Fig. 4).
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7.2 General salary cap An alternative, pro-CB rule may be the imposition of an identical ceiling on each 2 team’s total talent cost. This may be formalized as: C ¼ k R1 þR 2 ; that is, the cap is a 33 fixed share k [ (0, 1 ) of average total revenue. The rule is that no club can spend more than C for talent. It is natural to assume that the constraint is large enough to be not binding for the small, PM team. Alternatively, we might adopt a new approach in behavioral economics called asymmetric paternalism approach (Camerer et al. 2003) and argue that k be set such as to leave the rational PM team unaffected and bind only the WM team which spends too much on talent due to the excessive, not-fully-rational weight placed on present wins with respect to future financial viability.34 If the salary cap only affects the behavior of the big, WM team, the break-even condition (12) is replaced by the new condition: C C ¼ cx1 ) ¼ c x1
ð33Þ
which is the equation of a hyperbole. The effect of a general salary cap is to improve the league’s CB. As is showed in the following figure, the new equilibrium win percentage for team 1 is lower than that at the original mixed case equilibrium. This because the downward sloping GSC curve, which by assumption is below AR1, necessarily crosses the MR2 curve on the left of point E. Moreover, the equilibrium level of talent cost diminishes, as an effect of the reduced competitive pressure on the talent market’s demand side. Finally, note that if the cap is binding for both teams, the equilibrium is at point K, with perfect CB (each teams wins 1/2 of its games) (Fig. 5). Formally, by replacing (12) with (33), we obtain the solution: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b2 m2 ðm2 2b2 Þ2 þ 8b2 C GSC x1 ¼ ð34Þ 4b2 As before, let W = m2 – 2b2, so that the term under the square root may be written as W2 þ 8b2 C and the result of the square root as W + D, with D[ 0. The solution then is: xGSC ¼ 1
2b2 m2 ðW þ DÞ 4b2
ð35Þ
We may use (35) to prove, first, that only one of the solutions is acceptable. In fact, 2 ðWþDÞ 2 D ¼ 2b2 m4b ¼ 4b2 2m is always negative due to the minus solution xGSC 1 4b2 2 33
In US professional sports, the share k is bargained by the players’ and the club owners’ associations.
34
This, of course, only in case the WM behavior depends on a less-than-fully rational evaluation of present and future alternatives. Such is not the case when the owner rationally chooses to be WM because, say, this improves her non-sports business. On the limits of a general cap with respect to asymmetric paternalism, also see below in this section.
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AR1 MR1
AR2 MR2 AR1
GSC MR2 E
cˆ K c GSC
EGSC
GSC
X2 = 1
1/2 X1
cˆ
c GSC
x1
X1 = 1
Fig. 5 Effect on the ‘‘mixed’’ case equilibrium of a general salary cap
2 þðWþDÞ W + D [ 2b2 – m2. Hence, only the plus solution xGSC ¼ 2b2 m4b ¼ 4bD2 is 1 2 meaningful. Second, by repeating the reasoning of the revenue-sharing case, it may be showed that x~1 ; as given by (21), is always larger than xGSC , so that the policy 1 measure is effective in improving the league’s CB. Yet, it has been argued in the TSE literature that a general salary cap suffers from two problems: an enforcement problem, when the cap is binding on a PM team which is forced not to equate its marginal revenue and marginal cost, and an efficiency problem, since under this policy measure the league’s total revenues are not maximized.35 Moreover, it may be added that the imposition of a general salary cap seems highly unlikely in the present environment of European team sports. This because one of the latter’s key features is the multiplicity of competitions to which at least some of the clubs take part. Some of these competitions (league, national cup) are purely domestic, but others, such as the UEFA Champions League, involve teams from all European countries and warrant a very high return under both economic and sporting terms to the best performers. Hence, were a domestic league to impose a general salary cap on its clubs, it would inevitably put them at a disadvantage in these pan-European competitions against teams from countries where no such measure is applied. In an open talent market, the best players would in fact migrate to the leagues where there is no constraint to the salary they may receive, thereby depleting the stock of talent in the league which adopted the cap. The only way to avoid this outflow of talent would be the imposition of an identical cap in all (or most) European leagues of a given sport at the same time: a difficult task given the probably different degrees of CB across the various countries.
35
See, respectively, Fort and Quirk (1995) and Kesenne (2000a).
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7.3 Team-specific salary cap Due to the above-mentioned difficulties, we may look for an alternative form of salary cap. This time the constraint is team-specific, i.e., no team can spend on talent more than a fixed share of its revenues: Ci ¼ bRi : A standard result in the literature is that such measure is ineffective in improving the CB. Yet, this not so in our mixed model. Start again by assuming that the cap is only binding for the big, WM team. The new break-even condition for team 1 is obtained by shifting its AR function: bR1 ¼ cx1 ) c ¼ b
R1 ¼ bðm1 b1 x1 Þ x1
ð36Þ
Solving the system gives the following results: bm1 m2 þ 2b2 bb1 þ 2b2
ð37Þ
2m1 b2 þ b1 m2 2b1 b2 b1 þ 2=b b2
ð38Þ
¼ xSSC 1 and cSSC ¼
\ x~1 .36 The diagram below shows It is immediate to verify that cSSC \ ^c~ and xSSC 1 the new equilibrium (Fig. 6). Therefore, the imposition of a team-specific cap is effective in improving the league’s CB and lowering the equilibrium level of talent cost, provided the cap is only binding for the big teams: something that is quite likely to obtain given the postulated difference in the teams’ goal and behavior. That such a measure is able to affect the CB in our ‘‘mixed’’ model is indeed far from surprising, given that the standard ineffectiveness result crucially depends on the assumption of identical behavior of the two teams. It may be surmised that a club-specific cap would be easier to implement than a general one even at a pan-European level. To require a club not to spend for players more than a fixed share of its own revenues is in fact a rather commonsensical rule of good management which should raise much fewer objections than the imposition of an arbitrary, one-size-fit-all ceiling on talent expenditures. The previous remark naturally leads to figure out a possible ranking of the suggested policy measures. Several dimensions are available for such a ranking, including of course the distance between the policy outcome and the ideal case of perfect CB. What is suggested here is that the three measures may also be classified according to either their actual enforceability or their consistency with the asymmetric paternalism approach recently advocated by behavioral economists. In both cases, it turns out that the club-specific salary cap is the best policy measure. First, because it is surely simpler to implement and enforce than both market-related 36
The second inequality holds provided the equilibrium cost is positive.
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AR1 MR1
AR2 MR2 AR1
AR1SSC MR2 E
cˆ
cˆ
ESSC c SSC
x2=1
c SSC
SSC
x1
xˆ1
x1=1
Fig. 6 Effect on the ‘‘mixed’’ case equilibrium of a club-specific salary cap
revenue sharing (which refers to a vague, multi-dimensional notion such as market size) or a general cap (which, as already said, suffers from enforcement and efficiency problems). Second, because if the key principle of asymmetric paternalism is to devise a policy capable of constraining the behavior of only those agents who are not fully rational, while leaving unaffected the other fully rational ones (Camerer et al. 2003, p. 1212), then a club-specific cap is the only measure which may always conform to this principle, while both market-related revenue sharing and a general cap may sometimes violate it (when, respectively, a big team is PM or a PM team spends more than C for playing talent).
8 Conclusion The basic idea behind the paper is that a more realistic characterization of European professional leagues is that of ‘‘mixed’’ leagues, namely, of an environment where, while some teams still pursue the traditional win-maximizing goal, some others modify their behavior in the direction of straightforward profit-maximization. In particular, it is argued that this characterization seems to best capture the recent development in the Italian Serie A where the financial imbalances caused to WM teams by sky-rocketing players’ costs have led a few clubs to embrace a profitoriented attitude. In the paper we show the impact that such an alleged ‘‘Americanization’’ of at least some clubs’ behavior should have caused to a league’s CB and we evaluate the effect of alternative, pro-CB policy measures. Analytically, we use a modified version of the standard 2- and 3-team EH–Q–F model, with teams pursuing different goals. The peculiarity of our exercise is that,
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Competitive balance in football leagues
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for reasons that are detailed in the paper, the club which is assumed to change its behavior in the direction of profit maximization is the small or medium one (respectively, in the 2-team and 3-team case). The outcomes are straightforward: a worsening of the CB in favor of the big club and a reduction of the equilibrium level of talent cost in the 2-team model; an increase of the big and, even more so, small team’s win percentages coupled with a reduction of the talent cost in the 3-team setup. These are results that may be easily compared with the data of the last decade or so of the Italian Serie A. As to the policy measures which may be adopted to improve a league’s CB, the paper focuses on three options: (1) market-size-based revenue sharing, (2) general salary cap, (3) team-specific salary cap. Remarkably, all are shown to be effective in promoting a more balanced competition in a 2-team ‘‘mixed’’ league. This goes against the classic invariance proposition according to which, generally speaking, no rule aiming at reducing the gap between rich and poor clubs is really capable of affecting the league’s equilibrium. However, it must be kept in mind that the suggested measures suffer from well-known limits, such as the difficulty of managing any rule, like the salary cap, which is not self-enforcing, or the possible existence of an outside opportunity for the best players, viz., a foreign league where they can earn salaries well above the domestic cap. In this respect, it is suggested that the most preferable measure should be that which is easiest to specify and enforce. Alternatively, the preference should be given to the policy capable of correcting the ‘‘irrational’’ exuberance, as far as talent expenditure is concerned, of WM owners, while leaving unaffected the behavior of PM owners. It is a nice feature that in both cases the best measure turns out to be a team-specific salary cap. In any case, the thrust of the paper goes beyond these specific policy measures. The main message is that team sports economic models should always explicitly deal with the heterogeneity of the clubs’ goals and behavior characterizing European, much more than American, leagues. This is a feature which depends on the different historical evolution and institutional design of professional sports in the two continents (Cain and Haddock 2005) and that should therefore never be ignored when one tries to devise a league’s new ‘‘rules of the game’’. Acknowledgments I thank Raul Caruso and an anonymous referee for their useful suggestions. The usual disclaimers apply.
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