The financial situation of the football clubs in the Belgian Jupiler league: are players overpaid in a win-maximization league?

Kesenne, Stefan

Introduction

Many of the 18 clubs in Belgian's first-division Jupiler league are facing serious financial problems; five clubs are losing money season after season or have a negative net-wealth, with liabilities larger than assets. Most financial analysts seem to agree that most clubs are spending too much money on transfer fees and player salaries. The question is if Belgian football players are overpaid, given also their poor performances, nationally and internationally. The general public impression is that Belgian football players are featherbedded with too much money and too much leisure with very few hours of training per week. If the objective of the Belgian football clubs is to win as many games as possible, rather than maximizing season profits, it can be shown, using simple economic theory, that players are better-off in a win-maximization league than in a profit-maximization league, because the demand for talent by a win-maximizing club is higher so that, given the supply of talent, the market clearing salary level in a competitive player market will be higher, too (see Kesenne, 1996, 2007; Fort & Quirk, 2004). If players in a competitive player market under profit maximization are paid according to their marginal revenue (their contribution to the team's revenue), does it mean that players in a win-maximization league are overpaid, that is: above marginal revenue? The next section shows that, based on a general team sports model, players in a win-maximization league are not, in general, overpaid. In the following section, starting from a more specific model, we try to derive the condition for overpayment in a win-maximization league. In the subsequent section, this condition is applied to the Belgian first-division (Jupiler league) clubs, which shows that, in general, Belgian football clubs tend to overpay their players or to hire talents beyond their means. The conclusion then follows.

The General Model

Assume that the season revenue of a club can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where revenue is positively affected by the market size m and the winning percentage w, but concave in the winning percentage, [t.sub.j] are the number of talents in the team. The season cost function can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where c is the market clearing unit cost of talent and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the fixed capital cost.

Because the player market in European football is liberalized after the Bosman verdict in 1995, the supply of talent in Belgian football is elastic. So the appropriate model to start from should be the non-cooperative Nash equilibrium model using Nash conjectures (see Szymanski, 2004). However, if the number of teams in a league is large enough, 18 in the Jupiler league, the non-cooperative Nash equilibrium approaches the Competitive Walrasian equilibrium using fixed-supply conjectures.

Each club has a certain wage/turnover ratio, equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Maximizing the winning percentage under the club's budget constraint:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a fixed amount of season profits that club management wants to guarantee, which is below maximum profits.

The first-order conditions which can be derived from the Lagrange function, with multiplier [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], can then be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

From the budget constraint, one can derive that the demand for talent in a win-maximizing team is given by the net-average revenue curve of talent (NAR), which is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Proposition:

The market-clearing unit cost of talent in a win maximization league [c.sub.w] can be below or above the marginal revenue of talent, depending on the size of the wage-turnover ratio.

Proof.

By definition, it holds that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking the first derivative w.r.t. talent, we find that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be positive or negative, it follows that the equilibrium unit cost of talent under win maximization [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be above or below the marginal revenue [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], depending on the size of the wage- turnover ratio [??]. If [T??] is low (that is: if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is high enough), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], can be positive, so that the MR is above the unit cost of talent.

So, in general, players are, on average, not overpaid in a win maximization league.

A Specific Quadratic Model

Given a general lack of available data on player salaries and their individual contributions to club revenue, we have to find another way to compare salaries and marginal revenues. One possible way out, in my opinion, is to compare the equilibrium unit cost of talent in a win maximization league with the unit cost of talent in a profit maximization league, where playing talent is assumed to be paid in accordance to its marginal revenue.

Starting from the following well-behaved linear marginal revenue function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

This specification is well-behaved because the marginal revenue of a win approaches zero if the winning percentage approaches 100%.

In a simplified 2-team Fort and Quirk (1992) model, under the Walrasian fixed-supply conjectures, one can derive that the marking equilibrium from the equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Resulting in:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The market-clearing unit cost of a talent in a profit maximization league [c.sub.p] can then be found by substituting (9) into the (8):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The same way, and taking into account that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the unit cost of talent under win maximization [c.sub.H], can then be found to be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where [bar.r] is the average wage/turnover ratio in the league. This unit cost of talent will be higher than the unit cost in a profit-maximization league if the average wage turnover ratio in the league [bar.r] > 1/1.5 = 0.67.

Belgian's Jupiler League

We have collected data on the clubs' season budgets and payrolls and have calculated the average wage/turnover ratio in the league (see Table 1). These data are taken from newspaper articles and from the internet; they cannot be considered to be very reliable, because these sources often provide different figures.

Looking at the wage/turnover ration of 12 first-division clubs in the Belgian Jupiler league in Table 1, we observe that the average wage/turnover ratio is almost 70%. We can conclude that players in the Belgian Jupiler League are on average paid above their marginal productivity. Obviously, the teams with very high wage/turnover ratios (above 90%) strongly contribute to this average overpayment of talent. These teams have hired too many talents, given their available budgets.

On top of that, one should also take into account that most Belgian football clubs are subsidized by the government. So, if the cash subsidies are counted as part of the budget or the turnover of the club, the calculated wage/turnover ratios in Table 1 are underestimated. There are also many hidden subsidies to Belgian football clubs; in most cases, the local governments are the owners of the stadium and/or the grounds. Also, the central government grants several tax exemptions and favorable social security contributions to football clubs. The taxpayer also pays for police protection against hooliganism (see Kesenne et al., 2007).

Conclusion

In this contribution, we have shown that there are good reasons to believe that players in a win-maximization sports league can, on average, be paid above marginal productivity. Given the high average wage/turnover ratio and subsidies in the Belgian Jupiler League, one can conclude that the public impression that the Belgian professional football players are overpaid is correct.

References

De boekhouding van de eersteklassers doorgelicht [The book-keeping of the first-division clubs]. (2008, May 29). De Standaard-Het Nieuwsblad.

Fort, R., & Quirk, J. (2004). Owner objectives and competitive balance. Journal of Sports Economics, 5(1), 20-32.

Kesenne, S. (1996). League management in professional team sports with win maximizing clubs. European Journal for Sport Management, 2(2), 14-22.

Kesenne, S. (2007). The economic theory of professional team sports, An analytical treatment. Northampton, MA: E. Elgar.

Kesenne, S., Vanreusel, B., Van Langendonck, N., & Steens, G. (2007). Publieke geldstromen voor de sport in Vlaanderen [Public money flows in the sports sector in Flanders]. In G. Steens (Ed.), Naar een nieuwe bewegingscultuu, Sport, bewegen en gezondheid in Vlaanderen 2002-2006 (Vol. 2, pp. 143-158). [To a new movement culture; Sports, exercise and health in Flanders 2002-2006]. Antwerpen, Belgium: F&G Partners.

Quirk, J., & Fort, R. (1992). Pay dirt, the business of professional team sport. Princeton, NJ: Princeton University Press.

Szymanski, S. (2004). Professional team sports are only a game: The Walrasian Fixed-Supply Conjecture Model, contest-Nash equilibrium, and the invariance principle. Journal of Sports Economics, 5(2), 111-126.

Stefan Kesenne (1)

(1) University of Antwerp, city campus, and Catholic University of Leuven

Stefan Kesenne is a professor in the Economics Department at the University of Antwerp and in the Department of Human Kinesiology at Catholic University of Leuven. His research interests include sport economics and labor economics.

Tabel 1. Belgian Jupiler League, 2006/07

Team Turnover Wage/turnover
 (mill Euro) ratio(%)

Anderlecht 35 53
Club Brugge 20 76
Racing Genk 17.6 54
Standard 16 78
AA Gent 9.2 83
Germ. Beerschot 9 54
Moeskroen 8.5 95
Charlerloi 6.75 54
Zulte Waregem 5.7 65
Westerlo 4.75 69
Roeselare 4.5 69
Cercle Brugge 3.5 91
Average wage/turnover ratio: 69.5 %

Source: Newspaper "Standaard-Nieuwsblad" (2008).