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Building blocks for a disequilibrium model of a European team sports league.

Building Blocks for a Disequilibrium Model of a European Team Sports League (1)

A standard equilibrium model of a team sports league was first elaborated on in the context of North American professional team sports leagues assuming that teams are profit-maximizing (El Hodiri & Quirk, 1971; Fort & Quirk, 1995). When economists adapted this model to European team sports leagues, instead of assuming a team's profit maximization they focused on a team's win-maximizing objective function. They did not care enough about how irrelevant some crucial hypotheses such as price flexibility or a balanced budget constraint would become when a team was no longer a profit-maximizing, price-taking entity. They did not mind that managing a club without a profit-maximizing objective boils down to accepting that its budget constraint might not be hard with the consequence that a club in the red would survive to its financial deficit (2). Price rigidity and soft budget constraint do not fit with an equilibrium model and pave the way for either excess demand or excess supply in the markets for teams' inputs (labor, talents) and outputs (gate receipts, TV rights revenues). This article contends that a disequilibrium model should fit well with a European team sport (soccer) league where a number of teams actually run financial deficits.

Building a sophisticated and comprehensive disequilibrium model is beyond the scope of this article. The purpose of this article is to convince--or appeal to--model builders that they would be better off considering a switch from equilibrium to disequilibrium models when a team sports league encompasses teams that do not maximize profits. For this purpose it suffices to establish a few mathematical inequalities that make up the first building blocks of a required disequilibrium team sports league model.

The literature review in disequilibrium sports economics revealed no previous work on the topic (1). Beyond the standard equilibrium model of a European team sports league, other hypotheses and contrasting empirical evidence are pointed out while two varieties of disequilibrium models are distinguished (2). A first building block for a disequilibrium model of a European team sports league comprises a labor market for talent in excess demand (3). Dropping the hypothesis of a homogenous talent unit leads to integrating labor market segmentation for differently talented players (4). A second building block deals with a market for sport matches in short supply (5). Differentiating the demand side between fans and TV viewers and, for the latter's demand, introducing a trade-off between watching a free-to-air and a pay-per-view TV channel is a last step (6). Avenues for further research are sketched (7) before concluding (8).

Literature Review in Disequilibrium Sports Economics: An Empty Shell

A search of the sports economics literature found no article devoted to a disequilibrium team sport league model. A few sports markets in disequilibrium are surveyed in Andreff (2012), such as an excess demand for some sports events, possible excess demand or excess supply in the different markets for sporting goods, and an excess supply of some sport arenas and televised sports. For team sports leagues, the only departure from a standard equilibrium was made by two Australian mathematicians, Tuck and Whitten (2012). Their dynamic simulation model of a win-maximizing sports league allows for nonequilibrium solutions when the complex stochastic dynamic characteristics of a league provided teams with incentives to underperform and tank. However, properly speaking, this mathematical model is not an application of disequilibrium economics.

None of the famous disequilibrium economists (Clower, Leijonhuvfud, Barro (3), Grossman, Malinvaud, and others) seemed applicable to sports economics so there is no application of disequilibrium economics to sports thus far. Disequilibrium economics modeling started with a re-assessment of how much the standard general equilibrium model is relevant for a market economy with frictions and imperfections and grew into its standardized form in Benassy (1982, 1983). It eventually became the most relevant theory to understand those disequilibria that plagued centrally planned economies with Korna'i (1980). More recently Korna'i et al. (2003) generalized to all the situations where disequilibrium is a self-reproducing and self-sustained economic regime, in particular in those industries with nonprofit organizations. European soccer teams are a case in point as long as (it is assumed that) they behave as win-maximizing and not profit-maximizing entities.

Beyond the Standard Equilibrium Model of a European Team Sports League

A standard equilibrium model of a European team sports league was designed by Kesenne (1996, 2000) with win-maximizing teams, which is usually assumed an appropriate hypothesis in the context of European soccer. This model was elaborated further and compared with the standard model of North American leagues (Szymanski, 2003). In order to build up the European league variant of the standard model, Kesenne altered two assumptions:

a. Teams are win-maximizing but still wage-takers in the labor market for talent.

b. Therefore they recruit as much talent as possible within their budget constraints, a behavior that leads to an arms race for talent (Rosen & Sanderson, 2001).

The standard model of a European team sports league thus writes as

Max [t.sub.i] (1)

[R.sub.i] ([m.sub.i], [t.sub.i]) - s. [t.sub.i] - [c.sub.i.sup.0] = 0 (2)

with [t.sub.i] the quantity of talent in a team i, [R.sub.i] a team i's revenue function, [m.sub.i] the size of its local market, s the market unit cost of talent (or wage) and [c.sub.i.sup.0] a fixed cost (stadium, management).

Using the Lagrangian objective function, first order conditions are

1 + ([[lambda].sub.i] [[partial derivative][R.sub.i]/[partial derivative][t.sub.i]] - s) = 0 (3)

hence [RM.sub.i] = s - 1/[[lambda].sub.1] < s (4)

[R.sub.i] - s. [t.sub.i] = 0 (5)

From (4) marginal revenue of talent is lower than marginal cost. For a given unit talent cost, a team's demand for talent that maximizes its wins is bigger than if the team were profit-maximizing; in a European league, a team spends more on recruiting talent than in a North American league. From (5) a team's demand for talent is not given by its marginal revenue curve but by its average revenue curve (revenue per unit of talent):

[bar.R] = [R.sub.i] / [t.sub.i] = s.

Szymanski (2004) stressed that two other assumptions are at odds with European leagues:

c. In a post-Bosman global labor market for soccer players, free entry of players makes irrelevant the assumption of a fixed supply of talents adopted with North American leagues.

d. Assuming that teams are wage-takers is controversial given the heavy impact of some large clubs--Manchester United, Chelsea, FC Barcelona, Real Madrid, Milan AC, Juventus, PSG, and a few others--on wage and transfer fee determination.

Some alternative assumptions

Though Szymanski (2004) and Szymanski and Kesenne (2004) underlined various limitations introduced by the aforementioned assumptions in the standard equilibrium model of a team sports league, attention must be drawn to some alternative hypotheses. The first three of them are crucially required to allow writing a balanced team's budget constraint such as (2)

H1: Infinite flexibility must be assumed for the variations of either the quantity of talent [t.sub.i] or market wage s or both on the market for team i's input; the same assumption pertains to the team i's product market for it to reach equilibrium.

H2: Identical units of talent can always be marginally added or substituted for, which requires the assumption of an infinitely divisible and homogenous unit of talent. No differentiation between any two talent units or two players is taken into account even though, in the real world, journeymen players are not as much talented as superstars.

H3: All clubs are run in such a way as to exactly break even in accounting terms, i.e., they strictly stick to their budget constraint having simultaneously a zero profit and a zero deficit.

None of these hypotheses is required for a disequilibrium model. They are dropped in the following argument.

Beyond these three assumptions, a last means for team i to adjust to its budget constraint consists in augmenting its revenues [R.sub.i] by increasing its market size [m.sub.i], that is attracting more gate receipts/TV rights revenues on its output market. This may happen through either a flexible ticket price or TV subscription fee on the one hand and, on the other hand, an unrestricted variation in the number of ticket holders and pay-per-view TV viewers. Analyzing a team's trade-off between output price and quantity flexible variations implies a need to integrate the detailed functioning of a second (product) market in the model, which is rare in the literature about team sports leagues. A major exception is found in Kesenne (2007), who demonstrates that equilibrium ticket price, with price being assumed to be infinitely flexible, is higher in a league with win-maximizing teams than in a league with profit-maximizing teams. However, in this model, neither stadium attendance and gate receipts nor TV rights revenues depend on any consumer demand function that would differentiate fans, season ticket holders, occasional spectators, TV viewers, and couch potatoes who may have different behaviors in the face of a league supply of sport fixtures.

H4: A realistic hypothesis must be examined that fans and TV viewers don't have the same consumer demand function. This means that a sport show is a differentiated (and not homogenous) product, whereas the product market of a team sport league is segmented according to different varieties of a same sport show consumed either in attending a stadium or in switching on a TV set, a lap top, or a mobile phone.

Two varieties of disequilibrium models

In disequilibrium models, there are always two markets, one for input (labor) and one for output (product). Imagine that H1 above is dropped for the two markets, and then the whole economy--the league--is in a double disequilibrium regime with either excess demand or excess supply on both labor and product markets. Recall that price rigidity, whether in the form of a fixed price, a (monopoly/oligopoly) sticky price, or price inelasticity generates market disequilibrium and quantitative adjustment, i.e., rationing (Benassy, 1982). A disequilibrium model of a team sports league must theoretically be able to combine excess supply on one market with excess demand on another market, or excess supply or demand in both into one of the three possible theoretical disequilibrium economic regimes: repressed inflation, classical unemployment, or Keynesian unemployment (Table 1). In this variety of disequilibrium models, a la Benassy the focus is on price rigidity or inelasticity.

In a deregulated labor market, win- or utility-maximizing teams in European soccer leagues overbid and overspend for talents, run deficits season after season, and sink into debt crises (2.3 below) (4). Lasting deficit and debt suggest that teams enjoy a soft budget constraint in their maximizing calculation, which points out a second variety of disequilibrium models in which firms (here teams) can either attempt a hard ([greater than or equal to] 0) or enjoy a soft ([less than or equal to] 0) budget constraint, and not only a balanced budget (= 0) as in (2) above (5). Kornai (1980) has demonstrated that firms that enjoy a soft budget constraint take stock of it to form an excess demand for inputs, typical of a shortage economy. If all or most firms (teams) are run with a soft budget constraint, then the whole economy (league) functions all the time in disequilibrium. It never returns to equilibrium: there are always teams in the red.

Following Kornai's model, there are five circumstances (hypothesis H5) that ex ante secure a firm (team) to enjoy a soft budget constraint:

H5a: The team is price-maker on its input and/or output markets. If the team is a wage-taker, as in the standard equilibrium model, then its budget constraint is hard or balanced (never a deficit).

H5b: The team can influence taxation rules, avoid or postpone tax payment (tax arrears are tolerated by tax authorities). If the team cannot, then its budget constraint is hard.

H5c: The team can receive state or municipality grants or subsidies to cover current expenses or finance investment. If not, the team's budget constraint is hard.

H5d: The team can be granted credit by banks whatever its deficit, debt, and insolvency. If it does not find any credit due to its deficit, debt, or insolvency, its budget constraint is hard.

H5e: The team's investment does not depend on retained profits because it can find external finance for investing, including from fans or "sugar daddy" investors. When investment is strictly constrained by the amount of profits, a team's budget constraint is hard.

To conclude, when its budget constraint is soft, spending more than its revenues is not a matter of life and death for the firm (team), as Kornai states it, and it will survive its deficit. A high survival rate of soccer teams, even when they are in the red, would confirm that it is so.

Eventually, soft budget constraint and price inelasticity must be at the core of a disequilibrium team sports league model.

European soccer empirical evidence

A great deal of facts and empirical evidence in European soccer leagues provide some grounds to go beyond the standard equilibrium model since the aforementioned H1, H2, and H3 hypotheses do not seem to be actually fulfilled in the real world. More recently, differentiation in the consumer demand for soccer games successfully passed through econometric testing (H4) whereas H5 is more than likely to be relevant.

Hypothesis H1, as regards the team sports league's product market, is at odds with a demand for games inelastic to price or the fact that teams fix their price in the inelastic portion of their demand curve, as various studies have empirically tested it. References are many in the sports economics literature (Noll, 1974; Fort & Quirk, 1995; Coates & Harrison, 2005; Coates & Humphreys, 2007) and for European soccer leagues (Andreff, 1981; Bird, 1982; Dobson & Goddard, 1995; Falter & Perignon, 2000). In the European context, both league monopsony in the labor market and player unionization and collective bargaining, though less widespread than in North American leagues, fuel some stickiness in wage determination.

With regard to H2, assuming a homogenous talent unit is a rather abstract concept. How many homogenous talent units are embedded in Lionel Messi as compared, for instance, with Jean-Marc Bosman? The metrics to make it sure are missing. The managers of a real soccer team actually do not know what a talent unit is; they can only assess how much talent one player has as a bundle of several heterogeneous talents. Thus a team maximizes the number of heterogeneously talented units enshrined in live and entire players it recruits. Moreover, the terms of trade and wage determination are dramatically different for players who face a high demand for their talents--the superstars (6)--from those for journeymen players. The arms race observed in European soccer targets superstars in excess demand, not journeymen players. Each superstar holds an embedded specificity of his talents and a monopoly position on the supply side of the labor market, and thus benefits from a rent included in his wage (Borghans & Groot, 1998) and transfer fee, a quite different market situation from excess supply of sometime unemployed journeymen players. Segmentation of the soccer labor market has been observed by several authors (e.g., Bourg, 1983) (7).

A balanced budget constraint (H3) introduces a strong assumption that is not realistic compared with the current weak governance and bad financial performance of many European soccer teams and leagues (Andreff, 2007). According to UEFA (2011) data, 56% of all European top division soccer teams were taking losses in 2010. Being in the red is more the rule than the exception, which translates into leagues operating off their economic equilibrium. Storm and Nielsen (2012) underlined that a number of European professional soccer teams chronically operate on the brink of insolvency for over a decade or so without going out of business. Teams are promoted or relegated according to their sporting outcomes, but they almost never disappear from soccer business due to financial deficits. The teams' survival rate is very high (8) despite soccer business generating losses. Spanish soccer teams are used to spending more on player wages and transfers than their revenues earn, and exhibit a rising indebtedness (Barajas & Rodriguez, 2010). The French soccer league experimented quite more years in the red than in the black over a decade (Aglietta et al., 2008) even though deficits and debts were smaller than in English, Italian, and Spanish leagues. Even in the German league, professional soccer teams are not run as if they were compelled to break even since they are nonprofit associations (Vereins) committed to win as many games as possible for the fans, not to make profit.

A distinction between two varieties of sport show consumers (H4) has been successfully tested, with fans and season ticket holders on the one hand and, on the other hand, TV viewers, casual spectators, and couch potatoes (Simmons, 1996; Buraimo & Simmons, 2008). There is a fans' demand for attending games in stadiums, which is distinguished from a demand for televised games (Buraimo, 2008). Thus there are two products, i.e., two sport shows derived from the same game. The demand for seating close to the pitch is registered at stadium gates, while the demand for televised games is registered through audience records on public and free-to-air TV channels, and the number of subscribers to pay-per-view and commercial TV companies. Fans demand a maximum number of their favorite team games with an individual utility growing with as many attended games as possible, and with a wishful expectation of home team wins (9). TV viewers and couch potatoes rather demand uncertain and high quality games; thus they are attracted to more balanced games, showing a sufficiently high uncertainty of outcome (Forrest et al., 2005).

At first glance it is not a same disequilibrium that prevails in the fan and TV viewer markets. The first one may be either in excess demand--when fans are queuing at the gate and then scalping--or excess supply--when a game is played in a nearly empty stadium. Both happen in European soccer. It is more difficult to assess which disequilibrium affects the TV viewer market. A preliminary quantitative signal leans toward the assumption of an excess supply of nonfree televised sports overall (10), though it is not an academic proof.

Turning now to hypothesis H5, big soccer teams definitely influence wage determination in the market for talent (Szymanski, 2004)--they are wage-makers--and, though less so, a league and its teams are price-makers on their product market in particular when the league is pooling TV rights for sale (H5a). In European soccer, many teams fail to pay taxes without being liquidated (H5b), and when they enter administration the reconstructed tax arrears are almost always among the debt obligations that are not being met (Storm & Nielsen, 2012). A number of soccer teams rent sports facilities to a municipality at subsidized rates (H5c). The enforcement of EU competition policy has restricted the possibility of municipal subsidies to professional soccer teams. But subsidization still occasionally happens, like in the Italian Calcio, with the government having stepped in league financing (the salve calcio state plan in 2002; Baroncelli & Lago, 2006). Catalan banks grant credits (H5d) to FC Barcelona, and Castillan banks to Real Madrid whatever their balance sheets (Ascari & Gagnepain, 2007). Teams often succeed in renegotiating and rescheduling repayments to the banks.

Plenty of facts validate H5e. Many European soccer teams are seen as too big to fail by their stakeholders, and they always find some institution (bank, TV channel, etc.) to bail them out or grant them a loan even though irrecoverable; soft subsidies are provided by sugar daddies or sponsors in the form of additional cash and capital in cases of looming insolvency (Storm & Nielsen, 2012). This is widespread enough to be under the scrutiny of the coming UEFA Financial Fair Play initiative (Franck, 2013). Some TV companies either take over some teams or finance a league again and again through increased TV rights from one broadcasting contract to the other (Andreff, 2011). Such money inflows allow overinvestment in talent at team level. German teams are used to gamble on success from which ensues a constant overinvestment in talents and players (Dietl & Frank, 2007), a characteristic of organizations with a soft budget constraint. Many European soccer teams purchase more inputs than they can afford with their revenues; they attempt to endlessly recruit more expensive players, and the richest clubs overbid each other to attract the best superstars.

First Building Block: A Labor Market for Player Talents in Excess Demand

The simplest disequilibrium model of a European team sports league comprising of win-maximizing teams that operate in a labor market with some wage rigidity, is

Max [t.sub.i] (6')

[R.sub.i] ([m.sub.i], [t.sub.i]) - s. [t.sub.i] - [c.sub.i.sup.0] [less than or equal to] or [greater than or equal to] 0 (7')

Assuming that the stadium and club management are not significant inputs of a team's revenue function, the fixed cost can be left out and then it follows

[R.sub.i] ([m.sub.i], [t.sub.i]) [less than or equal to] or [greater than or equal to] s. [t.sub.i] (8')

Due to wage rigidity, limited talent mobility across the teams will not be enough to trigger equality between marginal cost and average talent revenue--as in Equation (5)--while the market wage will not adjust to its equilibrium level. Thus, compared to equilibrium, team i will form either excess demand for talents--when the budget constraint (7') is [less than or equal to] 0--or a too short demand when it is [greater than or equal to] 0. In the first event, team i spends more on acquiring talents than it makes in revenues, which fits with the empirical evidence of the above-mentioned soccer leagues. In the second case, team i does not spend all its revenues on talent purchase; it is left with surplus (above normal) profit, and it demands a shorter quantity of inputs than in equilibrium. Now assume that team i meets a soft budget constraint, (6') to (8') transform into

Max [t.sub.i] (6)

[R.sub.i] ([m.sub.i], [t.sub.i]) - s. [t.sub.i] - [c.sub.i.sup.0] [less than or equal to] 0 (7)

and without fixed cost: [R.sub.i] ([m.sub.i], [t.sub.i]) [less than or equal to] s. [t.sub.i] (8)

If most teams behave this way, two logical consequences are a. the league itself repeatedly is in the red when aggregating the net financial results of all its teams (as soon as the profits of some teams are more than compensated by losses of most teams), and b. there is necessarily a permanent excess demand on the labor market for talents, triggering an endless arms race. Without a hard or balanced budget constraint, there is no brake on a growing demand for talent while the number of players talented enough to play in professional soccer leagues is limited, say to [T.sub.0]. The labor market for talents is in disequilibrium due to a teams' aggregated excess demand in the face of a limited supply of player talents to the league [T.sub.0]:

[n.summation over (i=1)] [t.sub.i] = T; T [greater than or equal to] [T.sub.0] (9),

where the number of teams in the league is n (i = 1, ..., n).

With an excess demand for talents, marginal revenue productivity (11) of labor [RM.sub.t] = [partial derivative][R.sub.i] ([m.sub.i], [t.sub.i])/[partial derivative][t.sub.i]] equalize marginal unit cost of labor when the last unit of t [T.sub.0] talents is recruited and the disequilibrium in the labor market for talents implies, [RM.sub.i] = [partial derivative][R.sub.i]([m.sub.i], [t.sub.i])/[partial derivative][t.sub.i]] [less than or equal to] s (10)

All the labor units, up to the last one recruited, are overpaid when excess demand prevails in the labor market. Due to their aggregate overall excess demand, teams are rationed by a short supply of talents and are eager to pay a salary much higher than marginal labor productivity of talent, i.e., to overpay players in order to attract them in a context of harsh competition across the teams on the demand side of the labor market (12). Then, recruited players provide a lower labor productivity than the salary they are paid for, which sounds like the exact opposite of Scully's sense of player exploitation (Scully, 1974). In European leagues with win-maximizing teams operating under a soft budget constraint, players are paid more than they would have been at equilibrium wage. Since all teams are embarked on an arms race to recruit players, namely the few available superstars, they accept to pay a wage higher than marginal revenue productivity of labor in order to outbid competing teams--in all European leagues after Bosman case. Paying more than equilibrium wage and recruiting less than the quantity of talents they demand, rationed teams are involved in an endless skyrocketing race of payroll increases, which is what is seen in European soccer in regards to superstars.

Another implication of excess demand in the labor market for player talents is that soccer teams with a soft budget constraint attempt to recruit too many players, although they cannot afford as many as they would have wished (demanded) due to the shortage of superstars. Teams spend their money without counting losses--and sometimes they cook the books to hide this reality--due to soft budget constraints. Moreover, operating on the demand side of an excess demand input market, teams are always scared of being short of inputs without being able to find one more superstar in the market--due to inequality (9). In such events, like enterprises in former centrally planned economies, teams hoard labor as a reaction to circumvent the consequences of operating on a shortage input market. In European soccer, the very existence of a reservation and transfer system until the Bosman ruling of 1995 enabled teams to keep their players. After Bosman case, teams recruited on their rosters more players than they really needed (13). Thus, there is some slack in each team; teams are overmanned--look at the rosters of various European soccer teams and the number of substitutes never used over a season. This slack is beneficial to players in terms of the relationship between wages and both working time (very few players play all the season games during 90 minutes)--i.e., work intensity--and labor productivity.

Heterogeneous Units of Talent: Labor Market Segmentation

Now let us introduce a qualitative differentiation of talents between the most talented superstars and less talented journeymen players. It was assumed above that the supply of player talents is limited. Is it as much likely to be true for journeymen players as for superstars? Obviously not since one can witness in all European top soccer leagues a number of journeymen players who are unemployed at the start of every season (14). Unemployment is a crucial index of excess supply. Such observation apparently contradicts inequality (9), which means excess demand on the labor market for player talents. Therefore, for a journeyman player, segment of the labor market excess supply is to be modelled instead of excess demand on a superstar segment. Beforehand, the next point is to be considered: Is a team's excess demand for superstars on one market segment compensated by excess supply of journeymen players on the other market segment? It is assumed here that there is not full compensation: excess demand for superstars exceeds excess supply of journeymen players and the inequality (9) still remains relevant for the labor market overall.

Let [T.sub.S] stand for the overall number of available superstars and [T.sub.a] the overall number of available journeymen players. It follows that

[T.sub.s] + [T.sub.a] = [T.sub.0] (15).

Now a team has to maximize an assortment of superstar and journeymen player talents in order to maximize its wins (16), and its soft budget constraint is to be rewritten in such a way as to take this assortment into account (17). Below [t.sub.si] is defined as the demand for superstars by team i, [t.sub.ai] the demand for journeymen players by team i, [s.sub.s] the market wage for superstars, and [s.sub.a] the market wage for journeymen players. Thus for team i:

Max ([t.sub.si] + [t.sub.ai]) (16)

under a soft budget constraint:

[R.sub.i] ([m.sub.i], [t.sub.si] + [t.sub.ai]) - [s.sub.s] x [t.sub.si] - [s.sub.a] x [t.sub.ai] [less than or equal to] 0 (17)

and

[n.summation over (i=1)] ([t.sub.st] + [t.sub.at]) [greater than or equal to] [T.sub.0] (18).

If, as assumed, excess demand for superstars more than compensates excess supply for higher journeymen players, the labor market disequilibrium in the superstar segment becomes

[n.summation over (i=1)] [t.sub.si] [greater than or equal to] [T.sub.s] (19)

In the superstar segment of the market, excess demand coincides with a wage higher than marginal revenue productivity of labor (20):

[RM.sub.si] = [partial derivative][R.sub.i]([m.sub.i], [t.sub.si])/[partial derivative][t.sub.i]] (20)

Superstars are not directly competing against each other, with every superstar being in a monopoly position over his/her practically nonsubstitutable specific talent. In the journeymen player segment of the market, excess supply of talents (21) drives market wage down lower than marginal revenue productivity of labor (22) for these lower-quality and more competing talents:

[n.summation over ([i=1])] [t.sub.ai] [less than or equal to] [T.sub.a] (21)

[RM.sub.ai] = [partial derivative][R.sub.i]([m.sub.i], [t.sub.ai])/[partial derivative][t.sub.i]] [less than or equal to] [S.sub.a] (22)

Inequality (22) shows that journeymen players are subject to exploitation in a Scully's sense; they are paid less than their marginal productivity. They suffer from being in excess supply as well as from the monopsonistic situation of the league (15) (coordinated team owners) in the labor market; thus they bear a rent levied by owners on their salaries. An opposite asymmetry prevails on the superstar segment where the league's monopsony is countervailed by a strong monopoly situation of each superstar due to the uniqueness of his talents, skills, reputation, performances, record of achievements, and so on, and his or her absolute exclusivity over them.

A disequilibrium model of a league with win-maximizing teams operating under soft budget constraints in a segmented labor market describes an arms race for superstar talents fuelled by excess demand, superstars' skyrocketing wages (higher than their marginal labor productivity) that trigger teams' payroll overruns, the unemployment of journeymen players in excess supply used as a safety valve or an adjustment variable by team owners, and their lower wages paid at a rate below their marginal labor productivity.

Second Building Block: A Market for Sport Shows in Short Supply

Let us assume for a while that there is no difference between team fans attending a game at a stadium and TV viewers. Thus, on the demand side are found only homogenous fans who basically demand attending as many games of their favorite team i as they can financially afford (16); [d.sub.hi] stands for a fan h demand of team i games. A fan's utility is assumed to increase in strict proportion to the number of games attended. Since a European top soccer league is a cartel of teams supplying professional soccer games in a country, it is in a monopoly position to fix its output supply, i.e., the number of fixtures once given the number of teams in the league; [n.sub.i] stands for all the team i games supplied by the league over a season. This monopoly supply creates a game shortage on the supply side of the market for soccer matches and enables teams to maintain a price high enough to make a profit that includes a monopoly rent.

Consider one fan h: as any consumer, her demand function of team i games depends on her initial money endowment [m.sub.0h] (savings, assets), her income share available for paying tickets at the gate Inch (17), and the fixed ticket price of a game [bar.p] (18). First, imagine that there was just one fan per team i; her behavior could be described as

Max [d.sub.i] (23)

[d.sub.i] ([m.sub.0], Inc, [bar.p]) [greater than or equal to] [n.sub.i] (24)

[bar.p] > 0 (25).

In European soccer, each team plays twice against all other teams, one home and one away game, so that the supply of team i games is [n.sub.i] = 2 (n - 1) while the overall league supply of games over a season is 2 n (n - 1). This is total supply of professional soccer games in a country since creating another top soccer league in the same country is prohibited. Stadium capacity is assumed to be constant over a season and then the supply side overall is fixed by multiplying all stadiums' capacities by the overall number of games 2 n (n - 1).

Now assume that the overall number of a team sport league fans is g (h = 1, ..., g) and the number of team i's fans is [g.sub.i] ([h.sub.i] = 1, ..., [g.sub.i]), with [less than or equal to] g. All team i's fans are subject to the same constraint, a shortage in the supply by the league of their favorite team games, which number is restricted to [n.sub.i] = 2 (n - 1). Thus, for any team i's fan h, maximizing utility boils down to

Max [d.sub.hi] (26)

[d.sub.hi] ([m.sub.0h], [Inc.sub.h], [bar.p]) [greater than or equal to] 2 (n - 1) (27)

[bar.p] > 0 (28).

Whatever fans' expenditures over a season, the number of games they actually attend cannot be bigger than the number of their favorite team i games supplied by the league: 2 (n - 1). They are rationed by the league to this maximum number of team i's games over the whole season (19). Inequality (27) expresses the potential excess fan demand facing the league's short supply. With a fixed price the market will adjust in quantity, which is known as market rationing; fans adjust to available supply of games at a given price. The short side of the market, i.e., the league's supply of games, rations the longer side, i.e., fans' demand. Moreover, since fans' demand is trivially inelastic to a game fixed price, the price variable can be dropped, as in most disequilibrium models with price rigidity. Then the constraint (27) simplifies to

[d.sub.hi] ([m.sub.0h], [Inc.sub.h]) [greater than or equal to] 2 (n - 1) (29).

At the level of a whole league, that is for all games and all fans, market disequilibrium clearly is short supply (shortage) of games compared to fans' demand--from their standpoint, they are in excess demand, written as

[n.summation over (i=1)] [[g.sub.i].summation over (h=1)] [d.sub.hi] ([m.sub.0h], [Inc.sub.h]) [greater than or equal to] 2n (n - 1) (30).

At first sight, it could be objected that fans' demand does not take into account some variable reflecting game quality. There are various candidates for a game quality variable such as the team i's standing in the championship, its win percentage, its quantity of talents [t.sub.i] or its quotation by bookmakers. With regards to overall market disequilibrium (30), one could think of introducing some index of competitive balance on the left-hand side demand function [d.sub.hi]. However, this variable is not considered here under the assumption that a genuine team i's fan is not attracted to the stadium by game quality, the opponent team quality, or his or her favorite team quality. The typical fan attends simply because this is the emotional and usual fan behavior whatever their favorite team standing, win percentage, competitive balance, and so on. This assumption will be dropped further when other game spectators, such as TV viewers or couch potatoes, are differentiated from fans.

Finally, relations (6) to (22) and (26) to (30) describe two building blocks of a team sport league's economy that exhibits a double excess demand in labor and product markets, which is known to reproduce a repressed inflation regime (Benassy, 1983), also coined a shortage economy by Korna'i (1980), to which labor market segmentation has been added. As long as the budget constraint on soccer teams is soft--i.e., the breakeven point is not actually reached or enforced as a permanent governance rule--European soccer leagues will remain in such a regime. The arms race among the clubs to acquire the most talented players in the market will go on. Fans will go on being frustrated by waiting (queuing) for more games of their favorite team since their willingness to attend and pay is bigger than the number of their favorite team games fixed by the league.

Differentiating between Fans and TV Viewers Then between Free-to-Air and Pay-per-View TV Channels: A Product Market Differentiation

Following the distinction between two sorts of sport fans that emerged in the sports economics literature (European soccer empirical evidence, above), it is assumed that fans and TV viewers do not behave the same way in the market for sports matches. The latter is to be split into two segments for two differentiated products. As regard fans' demand and the corresponding segment of the market for sport shows, nothing is changed compared to above inequalities (26) to (30). However, a second segment of the market for televised sport shows must be introduced into the model together with a consumer demand function for televised games by TV viewers.

Assuming that TV viewers audience is attracted by outcome uncertainty and not by a specific team i games, their variable of interest is no longer the number of their favorite team i's games attended. One variable that encapsulates a game uncertainty of outcome lies in respective win percentages of two opponent teams i and j, that is [w.sub.i] and [w.sub.j]. Thus TV viewers' utility is maximized when the outcome uncertainty of a televised game is the highest, that is, both win percentages of the two teams are the highest; their utility function is assumed to increase in win percentages. It is only beyond a given threshold of the couple ([[bar.w].sub.i], [[bar.w].sub.j]) that a game is considered to be attractive enough to represent a high quality product worth watching. Beyond this threshold, that is, for , viewers switch on their TV, otherwise they do not. It follows that TV viewers' demand is high for high quality games, while only a proportion of overall games supplied by the league (assumed all to be televised) over a season will pass the quality threshold and will be watched. Let [d.sub.kij] stand for the number of high quality games between a team i and an opponent j that a TV viewer k (k = 1, ..., r) demands. TV viewer utility is assumed to increase in strict proportion with the number of high quality games watched. Assume also that TV viewers of the k vintage only like watching televised games for free and address their demand exclusively on free-to-air TV channels. Since not all the games televised by these channels are high quality, i.e., passing the threshold, TV viewers k are rationed. If, say, only a proportion [lambda] (0 [less than or equal to] [lambda] [less than or equal to] 1) of all televised games on free-to-air channels are high quality, then it follows for any TV viewer k:

Max [d.sub.kij] (31)

[d.sub.kij] ([m.sub.0k], [Inc.sub.k], [bar.w]) [greater than or equal to] 2 n (n - 1) (32)

p = 0 (33).

Overall market excess demand derives from aggregating all the individual TV viewers' demands:

[r.summation over (k=1)] [d.sub.kij] ([m.sub.0k], [Inc.sub.k], [bar.w]) [greater than or equal to] 2[lambda]n(n - 1) (34).

This was the observed real situation when there was just one public free-to-air monopoly TV channel in operation per country. It may still be so now, when there are a few coordinated or collusive oligopolistic public and private TV companies. In practice, this translates into high audience ratios (or high TV market shares) when the 2 [lambda] n (n-1) high quality games, with high outcome uncertainty, are broadcast on the one hand. On the other hand, only low audience is reached for a number n (1 - [lambda]) of boring games without enough outcome uncertainty. Relation (34) paves the way for empirical studies about audiences of televised sport on free-to-air channels in order to check whether they are low or not.

Now consider that televised games are no longer broadcast for free by public and free-to-air but by pay-per-view and commercial TV channels at a subscription fee [p.sup.*]. The market situation (equilibrium or disequilibrium) will crucially depend on the level of the subscription fee--fixed ahead for all over the season--by TV companies in oligopoly situation. Of course, TV viewers v (v = 1,..., z)--those willing to pay for televised sport--will take [p.sup.*] into account in their demand function [d.sub.vij] when maximizing their utility so that

Max [d.sub.vij] (35)

[d.sub.vij] ([m.sub.0v], [Inc.sub.v], [bar.w], [p.sup.*]) [greater than or equal to] or [less than or equal to] 2 [alpha] n (n - 1) (36)

[p.sup.*] > 0 (37)

with [alpha] the proportion (0 [less than or equal to] [alpha] [less than or equal to] 1) of all games televised on pay-per-view channels which are high quality and that TV viewers are willing to pay for.

If, by chance, a TV channel exactly fixed ex ante the single value of the subscription fee [p.sup.*] that transforms inequality (36) into an equality, then this TV company would have found the price that TV viewers are exactly willing to pay for a proportion a of all the broadcast games without spending more than they can afford given their savings [m.sub.0v] and incomes [Inc.sub.v]. In such circumstances, [p.sup.*] is the market equilibrium price. In the current functioning of TV channels that overbid to obtain the broadcast rights of European major soccer leagues, a TV channel that must cover the cost of such rights can find the equilibrium subscription fee only by chance. If a TV channel ex ante fixes a subscription fee lower than [p.sup.*], inequality (36) will be [greater than or equal to] and TV viewers will be rationed in high quality games but will pay a low subscription fee for this ration, lower than their willingness to pay. If, as it is more likely to happen with oligopolistic commercial TV companies, a TV channel ex ante fixes a subscription fee higher than [p.sup.*], a number of potential subscribers will not subscribe, inequality (36) will be [less than or equal to] and the sport (soccer) broadcast market will be in excess supply.

Avenues for Further Research

Beyond the two building blocks shown here, a great deal of work remains to be done to achieve a complete disequilibrium model of a European team sports league. The first task is to check for and analyze plausible interactions between the labor and product markets as the one suggested in endnote 16. Before doing so, a third building block should be elaborated on. Indeed, TV viewers are not supplied with televised sport fixtures directly by the league but through an intermediary market for sport TV broadcasts on the supply side of which stand either a monopolistic league that pools TV rights or teams that negotiate with TV channels on their own (20). On the demand side, oligopolistic competition (21) between TV channels does not usually translate into flexible prices (TV rights), all the more so when channels are facing a monopoly league and the bid-winning TV channel is cursed. Thus this intermediary market is likely to operate off equilibrium. Once completed with a third market, the disequilibrium model should be resolved, and its mathematical properties exhibited, a task that calls for help from skilled model builders. Then, before data collection, some indices should be designed in order to assess the existence and handle metrics of excess demand (e.g., outlying inflation for superstar wages, length of a fans waiting list, scalping) and excess supply (e.g., journeymen player unemployment, rates of stadium utilization, TV audience ratios). A further step, of course, should be robust econometric testing of the various building blocks after including appropriate disequilibrium indices (22).

In order to join a classical sports economics approach, at least to some extent, competitive balance should be introduced in the model, not only as a threshold to be reached as in (32), but also in the labor market block. Then competitive balance will be one of the interactive variables between two markets to be examined. An interesting question will emerge: Is a sport contest more--or less--balanced in a team sports league that is in economic equilibrium or disequilibrium? A more extensive comparison between the standard equilibrium and newly suggested disequilibrium models of a team sports league could be conducted with regard to mathematical properties, fitness to the empirical evidence of open leagues like in European soccer, and policy recommendations. A preliminary intuition derived from the first building block is that a salary cap would not moderate payroll growth as long as the labor market for superstars is in excess demand, that is, as long as the teams' budget constraints were not to be hardened up to not tolerating any financial deficit. From the disequilibrium model standpoint, the assessment of the UEFA Financial Fair Play, for example, should be different than the one derived from the standard equilibrium model. Once actually enforced, Financial Fair Play rules are likely to alleviate repressed inflation and somewhat curb superstar wage inflation in hardening the teams' budget constraints.

A last avenue for further research could consist in disentangling the three theoretical situations of excess demand (< 0), excess supply (> 0), and equilibrium (= 0) in each building block. Then, fragmenting each side of a market into subgroups of teams with different (budget) constraints, it would be possible to compare, within the same model, the effect of, for instance, a soft, a hard, and a balanced budget constraint. It would be of particular interest to allow deficit for a given subgroup of teams and impose the precondition of breaking even on other teams in the league, a differentiation that seems relevant to European soccer leagues. Then the model resolution will exhibit how the excess demand for superstars and competitive balance would evolve under such an assumption. Disentangling the three theoretical solutions fits here with the real world of some teams sticking to their budget constraint while others are overinvesting in talents, a major concern for policy makers in charge of European soccer today.

Conclusion

This article has explored two building blocks for a disequilibrium model of European team sports leagues in excess demand on both markets for talents and sport shows with win-maximizing teams. The suggested model presents two breakthroughs. For one, it fits well with a number of actual facts characterizing European soccer leagues such as price rigidity, teams' soft budget constraint, arms race for superstars, labor market segmentation, and a product market differentiation between fans and TV viewers. Second, with an infinite number of solutions, not only a unique equilibrium, it can adapt to different market situations and disequilibria in European soccer.

This first attempt opens several avenues for further research regarding how the building blocks presented here interact--and then a better model specification, how to design excess demand and excess supply empirical indices. We anticipate econometric testing that includes such indices, which can deepen a comparison between the respective strengths and weaknesses of the standard equilibrium and disequilibrium models of a European team sports league and, finally, can deliver appropriate policy recommendations.

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Endnotes

(1) I thank the three reviewers appointed by the IJSF for their comments; any remaining mistake is of my own.

(2) The article basically focuses on European professional soccer teams and leagues.

(3) With the anecdotal exception of Barro (2000).

(4) See the Journal of Sports Economics 7 (1) 2006 special issue on the financial crisis of European football and a part of the JSE 8 (6) 2007 issue.

(5) A disequilibrium model is more general than an equilibrium model which the solution is reached when the budget constraint exactly equals zero. In all the many (infinite number of) events when the budget constraint is either > 0 or < 0, disequilibrium self-sustains and lasts. A unique solution (= 0) is not crucially looked for in disequilibrium modelling where supply-demand equilibrium is considered as pure hazard or strictly theoretical. Korna'i (1980) compares Walras economic equilibrium with the point of absolute congealing in physics. Both have no empirical existence while being at the roots of theoretical reasoning and practical metrics.

(6) A superstar effect on wage determination has been clearly exhibited in Italian soccer by Lucifora and Simmons (2003).

(7) That a qualitative differentiation between player talents triggers labor market segmentation, though in American baseball, is also referred to in Hill and Spellman (1983) and Vrooman (1996) since there are two different wage rates for differently talented players in their models.

(8) A high survival rate is clearly verified in English soccer by Kuper and Szymanski (2009).

(9) Though tested for NBA basketball, fans wish at least two-thirds of their favorite team wins (Rascher and Solmes, 2007).

(10) In France in 2010, 98,000 hours of sports were televised by pay-per-view and commercial TV companies against 2,000 hours by public and free-to-air channels (CSA, 2011). A TV viewer willing to subscribe to all the supply of sports broadcasts would have been offered to pay over 2,000 [euro] per year. If addicted TV viewers can envisage spending such an amount for watching sports all day and all year long, nobody can spend more than 24 hours (98,000/365 = 268 hours) per day watching his/her TV screen.

(11) As one is used to define marginal productivity - since the entire model is not resolved so far.

(12) When a market is in disequilibrium, one side (for instance supply) of the market is shorter than the other one (demand); therefore the excess demand. In an excess supply situation, the demand side is shorter than the supply side. This means that, in an excess demand market, aggregating all the microeconomic demands (of all teams) comes out with a bigger quantity of talents than the (aggregated) quantity supplied by suppliers (all players). In excess supply, the aggregate quantity of supplied talents is bigger than the (aggregated) demand of all teams. Usually those economic agents on the short side of a market have a stronger bargaining power than those on the longer side; they successfully negotiate and bargain on their own terms--prices (thus they are price-makers) and transaction conditions -, and obtain a better pay off for what they deliver to the market.

(13) Since a soft budget constraint leads to hoard labor within the enterprise (team)--thus hedging against future labor market shortage- all European soccer teams are eager to recruit as many players as possible, including disposing of a great number of potential substitutes to seat on the touch-bench.

(14) Some of them, often not accounted for as unemployed, simply revise downwards the terms of their supply of talent and switch to a lower division team or a weaker foreign league. Supply (and demand) revision and switching from one market to another are at the heart of rationing schemes and adjustment processes in disequilibrium models.

(15) Though weaker in European soccer leagues than in North American major leagues.

(16) In addition, they wish, expect, or even bet on their favorite team winning as many games as possible (see endnote 9). However, they realistically cannot have a straightforward demand for home and away wins, due to outcome uncertainty. Their demand for wins is indirect and takes the form of their support (or request) to their team overinvesting in talents, which exactly fits with team behavior as described in inequalities 6 to 8. This is an interaction between the product and labor markets.

(17) Inch is the share of his/her overall income that a fan h can afford to pay for game attendance.

(18) This assumption materialises price rigidity in tune with the literature mentioned in 2.3 above.

(19) Here it is assumed that a fan wants to attend all his/her favorite team's games, home and away, and that he/she can financially afford it. Of course, it could easily be assumed that he/she wants attending only home games, then the supply he/she faces would simplify to (n - 1) without any major change in the model.

(20) Some insights are found in Andreff and Bourg (2006) and in Kesenne (2007, p. 22-25) for a tentative modelling in an equilibrium framework.

(21) Empirical evidence of a league's monopoly pooling TV rights facing oligopolistic TV companies in European soccer (Buraimo, 2006; Cave and Crandall, 2001) suggests that prices are not flexible; transactions are often plagued with overbidding, a winner's curse (Andreff, 2012), and a higher than equilibrium price.

(22) Just to give an example, derived from (21), [T.sub.a] - [n.summation over (i=1)] [t.sub.ai] is an (unemployment) index signaling an excess supply in the labor market for journeymen players.

Wladimir Andreff is Professor Emeritus at the University Paris 1, Pantheon Sorbonne, Honorary President of the European Sports Economics Association and International Association of Sports Economists, and former president of the French economic association.
Table 1: Economic Disequilibrium Regimes

                   Repressed        Classical
                   inflation       unemployment

Product market   excess demand    excess demand
Labor market     excess demand    excess supply

                   Keynesian          Fourth
                  unemployment       regime *

Product market   excess supply    excess supply
Labor market     excess supply    excess demand

* This regime is mathematically unstable and eventually
degenerates into repressed inflation (Benassy, 1982).
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