A payroll efficiency analysis of Europe’s top football leagues

1 A payroll efficiency analysis of Europe’s top football leagues G. Villa* and S. Lozano Escuela Superior de Ingenieros, University of Seville Camino de los Descubrimientos, s/n, 41092 Seville, Spain * Corresponding author: E-mail: [email protected] Abstract Top European football teams have large budgets mainly due to the high wage bills they pay to the players. Therefore, it would be interesting to determine if these football teams are paying inflated salaries considering the sports results that they obtain each season in both national and international competitions. This study focuses on the top five European football leagues (Italian, Spanish, English, French and German). We propose a novel non-convex, nonparametric metafrontier analysis approach to determine whether the football clubs are overpaying their players considering their sporting performance. Goals against in both national and international competitions are modeled as undesirable outputs. Each football team is benchmarked first within its own league and then against all five leagues. From this, apart from estimating the payroll efficiency of each team, the average efficiency of each of these five leagues can also be computed. An exhaustive analysis and discussion of the results is presented using data from three seasons (2020-23). Some “important” football clubs pay salaries that are not justified by their performance when compared with other, more modest, clubs that pay salaries more in line with the sports results obtained. Ligue 1 is the league that, on average, makes the most efficient use of their payroll, followed by the Bundesliga and, somewhat behind, La Liga and Serie A. The Premier League occupies the last position in terms of average payroll efficiency. Keywords: European football clubs; payroll efficiency; non-convex metafrontier analysis; data envelopment analysis This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Management Mathematics following peer review. The version of record G Villa, S Lozano (2024) A payroll efficiency analysis of Europe’s top football leagues, IMA Journal of Management Mathematics, dpae014, is available online at https://academic.oup.com/imaman/advance-article- abstract/doi/10.1093/imaman/dpae014/7680269?redirectedFrom=fulltext 2 1. Introduction Association football is becoming increasingly popular around the world, and consequently, many football teams manage huge amounts of money. For the first time six major teams from European football leagues are among the 15 most valuable sports teams in the world in 2021 (Forbes, 2021), and it is not unusual that a noticeable percentage of their budgets are intended to pay the high salaries of their players. Although, in the case of the best players, the salary is partly determined by their image and prestige (which can also contribute to increase the club's revenue via merchandising), the annual payroll paid by the club to their players should be assessed in line with the achievement of sports results. Relating what is paid to the players to the sports performance of the team can help determine the payroll efficiency of each team presents for a given season. In this paper, we consider the football clubs that belong to the top five European leagues (namely, Italian, Spanish, English, French and German). The payroll efficiency of each club is computed using a metafrontier analysis approach (O'Donnell et al. 2008) in which each team is also benchmarked against the clubs in its own league, which allows determining an average payroll efficiency for each league. For both the group frontier (GF) efficiency and the metafrontier (MF) efficiency, data envelopment analysis (DEA) will be used. DEA is a nonparametric methodology commonly used to assess the relative efficiency of a set of so-called decision making units (DMUs) DMUs are modelled as input-output systems. Inputs are assumed to be the less the better while outputs are considered to be the more the better. Hence, a DMU is said to be inefficient if it consumes more inputs than is required for the level of outputs it produces or if it produces less output than it could with the inputs it consumes. DEA is a data-driven approach that only requires the input and output data of the DMUs. There may exist uncertainty in the data (e.g., Lozano and Khezri 2023) and also some of the variables may correspond to undesirable outputs (e.g., Emrouznejad et al. 2023b). From that data DEA infers the production possibility set (a.k.a. the technology) that contains all the operating points that are deemed feasible. The non-dominated subset of the technology is called the efficient frontier. The efficiency assessment of a DMU consists in projecting a DMU onto this efficient frontier and determining an efficient target operating point that dominates the DMU and proves it inefficient. As mentioned before, if there are no feasible operating points that dominate a given DMU then it is efficient and the DMU is projected onto itself. The target of a DMU shows, by comparison with the own DMU, the input dimensions in which there are excess consumptions (a.k.a. input slacks) as well as the output 3 dimensions in which there are shortfalls (a.k.a. output slacks). Based on these input and output slacks an efficiency score can be assigned to the DMU. This efficiency score is a measure of the distance between the DMU and its target. The larger this distance, i.e., the larger the potential improvements, the lower the efficiency score. DEA cannot only assess the technical efficiency in a given period but also the productivity change over time measured through the Malmquist productivity index (e.g., Zhu et al. 2024). And it can not only be used to assess past performance but also for production planning (e.g., Kazemi and Galagedera 2023) or to assess potential gains from the mergers of production units (e.g., Soltanifar et al. 2023). In the payroll efficiency assessment approach presented in this paper, the DMUs are the football teams. The single input considered, the one we want to focus on, is the players’ salaries. According to Hall et al (2002), who analysed English football clubs, high payrolls lead to better performance. Hence, in our approach, the outputs correspond to the sports results of the team along the season, including national competitions (i.e., First League and National Cup) as well as, if applicable, international competitions (i.e. Europa League and Champions League). Specifically, the outputs include the number of matches won in national (respectively, international) competitions and the number of goals for and against in national (respectively, international) competitions. As regards considering wins as an output, note that Einolf (2004) argued that in a not revenue sharing business model like European football teams, winning is everything, and owners spend as much as they do to achieve victory. Unlike the number of matches won and the number of goals for, the number of goals against is an undesirable output. This type of outputs is sometimes treated as inputs because, the same as inputs, these variables are the less the better. The alternative used in this paper is to model undesirable outputs as jointly weakly disposable with the desirable outputs (see Kuosmanen 2005). In any case, in this application, the outputs, whether desirable or undesirable, are considered non-discretionary, i.e., it is assumed that DMU cannot improve them at will. In other words, the outputs are given. It is the input that is under the control of the DMU and, hence, an input orientation is considered. The idea is, therefore, to reduce the input as much as possible without worsening the level of outputs. Payroll efficiency implies paying the lowest amount of players’ salaries given the sports results obtained. Thus, if a team A obtains the same or better sports results than a team B and, in addition, team A pays lower salaries to its players than team B, then team A is more payroll efficient than team B and team B could reduce its payroll, at least to the level of team A (or even less if team A is dominated itself by 4 some other more efficient team). Contrary to some claims (Scully, 1995; Quirk and Fort, 1997) that top teams labelled as payroll inefficient may be acting rationally to maximize revenue, not wins; we believe that in football, in the long run, if a team does not obtain sporting successes its prestige will be compromised and therefore it will be forced to reduce its expenses (including the payroll). One can find many examples of teams from the major European leagues to which this has happened. The proposed approach is innovative both as regards its aim as the methodology used. The aim of this research is to assess the direct link between football teams’ payrolls and sports performance. This is a relevant and simple research question which, to the best of our knowledge, has not been addressed using efficiency analysis. Thus, it is clear that some teams have deep pockets and can hire the best players in the market. That higher payrolls lead to improved sports performance is not so clear though. Large budget teams are often in the top positions of their national leagues and participate in international competitions. However, so also do other more modest teams which can, therefore, be assessed as more payroll efficient. This simple reasoning is behind the proposed approach, i.e., the larger the salaries paid to the players the higher should the team performance in terms of sports results be. An input orientation is used, which means that inefficiency comes from potential input reductions and means overpaying the players. This issue has different economic, social and institutional implications. On one hand, excess spending on salaries indicates an unproductive use of scarce financial resources that can be freed for other purposes. Thus, more and more football teams are putting sustainability at the centre of their strategic goals and developing initiatives to achieve those goals. See, for example, UEFA’s football sustainability strategy (https://www.uefa.com/insideuefa/sustainability/strategy/) or the Forever Green project of Real Betis football club (https://forevergreen.es/?lang=en). Other social implications of the excess payrolls involve its contribution to income inequality. There is a part of society that resents the exorbitant salaries paid to some football players, salaries which some find unjustified. The question of the high salaries of some clubs is also related to the fairness of the competition when the disparities between the financial resources available to the different teams allows some of them to hire the best players and thus gain an advantage. This is a growing concern that has moved UEFA to impose a limit on clubs' spending on wages, 5 transfers and agents' fees to a certain percentage of their revenue, starting in season 2023-24 (Stone 2022). As regards the contributions of the paper, apart from focusing on the payroll efficiency, is the use of an original DEA approach that uses non-convex metafrontier analysis and weakly disposable undesirable outputs. Moreover, we have considered observations from the last three seasons and an intertemporal approach, which increases the discriminant power of the method. The reason for using a metafrontier approach is that, although there is a global market for football players, the situation as regards relative payroll levels differs between countries. Hence it makes sense to assume a specific group technology for each league and a metatechnology for the whole sample. The non-convexity of the latter is assumed on the grounds that this assumption is more conservative than the alternative. The structure of the paper is the following. In Section 2 we review the relevant literature. In Section 3, the proposed approach is formulated and explained. Section 4 presents and discusses the results obtained. Finally, Section 5 summarizes and concludes. 2. Literature review 2.1. Metafrontier analysis When the DMUs to be benchmarked are heterogeneous and belong to different categories or groups, a metafrontier DEA approach is recommended (O'Donnell et al. 2008). In this case, for each DMU, a group efficiency and a metafrontier efficiency can be computed. One projects the DMU onto the efficient frontier of its group technology defined by the DMUs in that group, while the other projects the DMU onto the efficient frontier of the metatechnology that results from considering all the DMUs. The group efficiency score is always greater than (or at least equal to) the metafrontier efficiency score. Actually, the ratio of both efficiency scores, called the Technology Gap Ratio (TGR) or Metatechnology Ratio (MTR) of the corresponding group, provides a measure of the relative performance of the group technology (as compared with the metatechnology). It is like comparing the best practices of the group with the overall best practices. Two ways, however, have been proposed to define the metatechnology (Kerstens et al. 2019). The first defines the metatechnology as the conventional DEA technology associated to with the whole set of DMUs, i.e., DMUs are first pooled into a single set of observations and the usual axiomatic derivation of the DEA technology from this overall set is carried out. This usually results in a convex 6 metatechnology (e.g., Oh and Lee 2010). The other way, which is the one considered in this paper, defines the metatechnology as the union of the group technologies. This generally results in a non-convex metatechnology and projecting onto its efficient frontier requires specific DEA models (Huang et al. 2013, Afsharian and Podinovski 2018, Afsharian et al. 2018, 2019). 2.2. DEA applications to football The literature on DEA applications to football is extensive. Several authors (e.g., Villa and Lozano 2016, 2019, Guzmán-Raja and Guzmán-Raja 2021) have compiled lists of the most relevant articles in this field, indicating the inputs and outputs considered and the DEA approach used. Below we refer to the studies that are most relevant to this research, which will facilitate the identification of the corresponding research gaps. In particular, we are interested in studies that use team wages, labour costs or operating costs as inputs as well as those that have carried out a metafrontier analysis. Thus, for example, Tiedemann et al. (2011) proposed a metafrontier approach to assess the performance of individual football players taking into account their playing position (defenders, midfielders, forwards) while Espitia-Escuer and Garcia-Cebrian (2020) used a metafrontier approach to assess the efficiency of football teams in different zones of the playing field (defenders, defensive midfielders, holding midfielders, attacking midfielders and forwards). Many are the DEA approaches to football team efficiency that consider the team payroll, the coach salary, the staff wages or the total wages as an input (e.g., Barros and García-del-Barrio 2011, Barros and Leach 2006, Haas 2003a, 2003b, Haas et al. 2004) but they also consider other inputs and they do not focus specifically on trying to reduce the players’ salaries. An output orientation aimed at increasing the outputs is usually applied. An exception is Ribeiro and Lima (2012), who study whether football clubs in the Portuguese First League were spending more money than they needed to. In that study, the comparison was carried out only within a single league and their approach considered just a single output, namely the rank in the First League. Guzmán-Raja and Guzmán-Raja (2021) also used an input orientation to assess economic and sport performance, considering turnover and points won as outputs and staff costs and other expenses as inputs. Pérez-González et al. (2022) proposed a NDEA model in which the total assets and operating expenses of the teams are used to obtain results in the social and sport activities that, in turn, determine their revenue generation. Villa and Lozano (2024) also used NDEA, considering team wages as a shared input consumed by the 7 national and international competition processes which feed the value creation stage that produces market value of the team. They used a non-oriented slacks-based inefficiency approach. 2.3. Research gaps and main contributions Our claim is that, as regards team wages efficiency, the European teams are heterogeneous and hence a metafrontier analysis is due. However, we did not find in the existing literature any studies that compare teams from different leagues using a metafrontier approach. Nor have previous studies attempted to specifically analyze the payroll efficiency of teams of different leagues given their sports performance in national and international competitions. As indicated above, some studies have considered team wages as inputs but they did not consider multiple leagues or they did not use an input orientation focusing on team wages reduction. In the next section, we present an approach that tries to cover these two research gaps. Therefore, apart from the novelty of its payroll efficiency assessment aim, the proposed approach uses state of the art DEA methodology, including non-convex metafrontier analysis and weakly disposable undesirable outputs. 3. Proposed approach Let us consider the teams from different national leagues as the DMUs. Figure 1 shows the inputs and outputs considered. There is a single input, the annual players’ payroll, and six outputs that represent the sport performance of the team in a certain season, namely the number of matches won, goals scored and goals against in both national and international competitions. Since we are interested in the evaluation of the input efficiency of teams, we will consider that the sport results are non-discretionary variables and model them as per Banker and Morey (1986). This means that each team will be benchmarked against those teams with equal or better sports results. Also, note that the number of goals against in both national and international competitions are treated as weakly disposable undesirable outputs. 8 Figure 1. Payroll efficiency inputs and outputs Before formulating the proposed DEA models, let us introduce the required notation. Data j index for football teams. 0 index for the football team whose efficiency is evaluated. g index for the national league to which DMU 0 belongs. set of football teams from league g. annual payroll of football team . amount of desirable output k of football team (index k refers to no. of matches won in national competitions, no. of matches won in international competitions, no. of goals for in national competitions, no. of goals for in international competitions). amount of undesirable output b of football team (index b refers to no. of goals against in national competitions, no. of goals against in international competitions). Decision variables g D g j PAY g jDÎ g kj y g jDÎ g bj z g jDÎ 9 intensity variables for football team j from league g that represents the part of team j`s output that remains active. intensity variables for football team j from league g that represents the part of team j’s output that is abated through scaling down of activity level. reduction of annual payroll. Let us formulate first the Group Frontier (GF) DEA model that allows benchmarking a given team 0 against the other teams in its group. This produces an estimate of how much the players’ salaries of a given team 0 can be reduced just by comparison within its league. This is not the actual payroll efficiency, which will be determined by solving the corresponding Metafrontier (MF) DEA, i.e., comparing a given team 0 not only against those in its group but against all the teams in all the groups. Therefore, note that in the model below, the index g denotes the group to which DMU 0 belongs and the set contains all the DMUs in this group. GF model (1) s.t. (2) (3) (4) (5) (6) This input-oriented Variable Returns to Scale (VRS) DEA model is based on the linearization proposed by Kuosmanen (2005), when undesirable outputs are weakly disposable. Model (1)- (6) seeks to minimize the annual payroll of the team 0 belonging to the league g, when both, g j l g j µ q g D GF 0Minq=q ( ) gg g g 0 jj j jD g PAY PAY Î l+µ £q× å gg g j kj k0 jD g yy k Î l³ " å g gg g j bj b0 jD zz b Î l£ " å ( ) gg jj jD g 1 Î l+µ = å gg g jj ,0gjD freelµ³ ""Î q 16 Figure 4. Average (for the three seasons) of the GF and MF efficiency scores for Serie A 17 As regards the Italian league, Serie A, it has a high average GF efficiency together with a low MF efficiency. The three teams with the largest MF efficiency scores are INT, NAP and ATA. Although none of the three was MF efficient in all three seasons, INT was MF efficient in 2021-22 and 2022-23 and very close in season 2020-21. Other teams that are also above the mean MF efficiency are MIL, ROM, JUV and LAZ. It is interesting that the teams with the largest budgets are also the most efficient, which means that they must have also obtained good sports results. Without entering in the specific situation of each team during this period, it is interesting to note that the lowest average GF efficiency scores are around 0.4, which implies a significant payroll excess, and correspond to GEN, CAG and PAR, none of which played in the league the whole three seasons (GEN and CAG played the first two seasons but were both relegated in the second while PAR played only in the first season and was relegated). Not surprisingly, as happens in the case of other leagues, the teams with the lowest GF efficiencies are also those with the lowest MF efficiency. Note, however, that the fact that these teams obtained poor sports results is not exactly the reason for these teams to be so MF inefficient. Actually, it is the fact that other teams, probably from other leagues, with much smaller payrolls obtained similar or even better sports results. As regards the Spanish league, La Liga, the average GF efficiency is slightly lower than that of Serie A but the average MF efficiency is somewhat higher. Interestingly, unlike in Serie A, here around half of the teams have an MF efficiency above the mean. Two teams, namely VIL and GIR, have average GF efficiency equal to one but with the difference that while VIL played in the league the three seasons GIR was promoted to the league just the last season (and, by the way, has ranked 10th). However, when these teams (and most other Spanish teams for that matter) are benchmarked with the other national leagues they do not fare so well, with much lower MF efficiency scores. This is reflected also in their relatively low TGR. SEV, with similar GF and MF scores, is an exception to this rule. In the case of the Premier league, there is just one MF efficient team (MCI) and two more (BRE and SHE) that have been GF efficient although while MCI played in the Premier in the three seasons, BRE only played in the last two seasons and SHE in the first one. Note that, although the average GF efficiency is more or less similar to those of the other leagues (it is even higher than that of Ligue 1), the average MF efficiency is lower than that of the other leagues, just below that of Serie A and much lower than Ligue 1 or Bundesliga. The TGR for these teams are also lower than for other leagues, indicating that this is the most payroll 18 Figure 5. Average (for the three seasons) of the GF and MF efficiency scores for La Liga 19 Figure 6. Average (for the three seasons) of the GF and MF efficiency scores for the Premier 20 inefficient of all leagues if we assess it from the assumed pure sports results perspective. And not only is the average low but also more than half of the teams have MF efficiency below the average. As shown in Figure 7, all the teams in the French league have almost the same GF and MF efficiency scores and hence a TGR close to 1. Interestingly, although its average MF efficiency is the highest of the five leagues, its average GF efficiency is not high. Actually, it is lower than that of the other leagues, which means that although the French league sets the standard of payroll efficiency among the five leagues under study, within the own French league there are many teams that are not GF efficient. Anyway, there are two teams that are used as MF reference units for a large number of teams of all five leagues. These are CF, which was MF efficient in the two seasons in which it has played in Ligue 1 (seasons 2021-22 and 2022-23) and TCF, which was MF efficient in the only season it has played in Ligue 1 (season 2022-23). These two teams are those that get “the most bang for the buck” in terms of sports results. Compared with these teams most other teams in League 1 and, in fact, in the other leagues, are assessed as payroll inefficient. Finally, in the case of the Bundesliga, it can be seen in Figure 8 the high correlation between the GF and MF efficiency scores. All in all, the average GF efficiency is high (higher than that of Ligue 1) as is also the average MF efficiency (second to Ligue 1). Only one team (BAY) was MF efficient the three seasons although there are also other teams that were MF efficient in some seasons (DOR and TSG in season 2020-21, RBL and SGE in season 2021- 22 and SCF and FCU in season 2022-23). Figure 9 shows the average GF and MF efficiency of the different leagues in each season (symbols and , the above the theta symbol representing the average for all the teams of the corresponding league). Note that, by definition, the MF efficiency is always lower than the GF efficiency and hence their respective averages must fall below or on the bisector line. The slope of the line connecting each point with the origin represents the TGR of each league. The closest to the bisector line (and the largest TGR) corresponds to Ligue 1, which has consistently performed, on average, very well in payroll efficiency compared with the other four leagues. Although with a somewhat larger variability, the second position in MF payroll efficiency goes to the Bundesliga. Of the other three leagues, Serie A has higher average GF than La Liga but this has higher average MF efficiency, which in the end, is what GF q  MF q   21 Figure 7. Average (for the three seasons) of the GF and MF efficiency scores for Ligue 1 22 Figure 8. Average (for the three seasons) of the GF and MF efficiency scores for the Bundesliga 23 counts. Note the increase in average MF efficiency of La Liga in the season 2022-23. Finally, the Premier not only has the lowest average GF and MF efficiency scores but also, they seem to have worsened every season. Finally, translating the computed inefficiency scores into monetary units, Table 1 shows the average estimated excess wages paid in each league in each season. It can be seen that the average excess wages of the different leagues are significant and persistent, with the largest amount corresponding to the Premier league (average excess wages of around 70 M€), followed by the Spanish and Italian league (average excess wages of around 40 and 35 M€, respectively). The French and German leagues have smaller average excess wages (around 20 M€). Note that this is the average amount that the teams in each of these leagues is overpaying their players and that, if saved, would relieve the financial strain of some of them and allow all of them to free those resources for other, more socially-responsible aims. League 2020-21 2021-22 2022-23 Average of the three seasons Serie A 32.20 37.22 31.00 33.47 La Liga 41.28 47.17 25.47 37.97 Premier 51.19 71.88 79.59 67.55 Ligue 1 12.85 22.07 18.63 17.85 Bundesliga 17.57 17.27 18.77 17.87 Table 1. Average excess annual payroll (in M€) in each season for each league 5. Managerial implications The implications of this study span three levels, namely that of each individual club, the national league level and the international level. For each individual club the results of the analysis can be used in two different ways: to assess past payroll performance and to determine future salaries for the team. However, it is unlikely that the potential salary savings identified by the model can be attained in a single season. Clubs typically sign multiple period contracts with their players, which means that individual players salary negotiations may have to wait until these are up for renewal. Our suggestion is for each club to link team wages more directly to sports results. This can be done negotiating at the start of each season realistic sport achievements goals with rewards in case these are exceeded but also with penalties in case they are not reached. 24 Although, as we have seen, the situation of the individual clubs differs, policy implications can also be drawn at the national level. Thus, for example, public scrutiny on the total wages paid at the team level should be enhanced through transparency and self-reporting mechanisms. Football clubs financial audits and sound management policies should also be implemented by national governments. Finally, national government could also enact regulations facilitating the different stakeholders to participate and have say in the financial management of football clubs, which can prevent overt cases of overspending. Finally, harmonizing the national legislation and levelling the playing field is the responsibility of the supranational institutions such as FIFA and UEFA. This includes monitoring team wage efficiency and detecting cases of excessive overspending, which can affect negatively not only the sustainability of the clubs’ finances but also the competitive balance in the different leagues and at the international level. On the other hand, these supranational institutions, and ultimately the courts (recall the Bosman ruling), are also in charge of regulating and monitoring the international football transfers market to avoid inefficiencies and minimize transaction costs. Thus, facilitating players whose talent and performance is not appropriately rewarded in its current club to be move to another club helps reduce overall relative wage inefficiency by better matching wages and performance. Note that, in the end, payroll inefficiency stems both from overpaying for underperformance and underpaying for overperformance. 25 Figure 9. Average (for each league) of the GF and MF efficiency scores in each season