The grouping of stars: An application to professional sports

International Journal of Industrial Organization 17 (1999) 1009–1027

The grouping of stars: An application to professional sports Mark R. Frascatore* Department of Economics and Finance, Clarkson University, Potsdam, NY 13699 -5785, USA Received 18 December 1997; received in revised form 31 December 1997; accepted 31 December 1997

Abstract This paper uses a two-period model to analyze competition for high-quality labor in a professional sports market. Leagues vertically differentiate their products by competing for the services of a small number of stars, thereby endogenizing the cost of product quality. If utility is linear or convex in the number of stars, one league successfully employs all of them. This grouping of stars results in an equilibrium market structure of either a monopoly or a duopoly, depending on the opportunity costs of players and the number of consumers. Market efficiency of the resulting equilibrium is then discussed.  1999 Elsevier Science B.V. All rights reserved. Keywords: Vertical product differentiation; Worker synergy; Market structure; Professional sports JEL classification: L15; L83

1. Introduction If labor is heterogeneous in quality, the grouping of the most talented workers to specific firms in an industry can serve to vertically differentiate products. The grouping of ‘‘star workers’’ is quite common in professional sports and other entertainment industries. Professional sports leagues tend to hire only the very best

* Tel.: 315 268 3850; fax: 315 268 3810. E-mail address: [email protected] (M.R. Frascatore) 0167-7187 / 99 / $ – see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 98 )00010-1

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athletes to compete against one another, and entry into these industries with star players has proven difficult. For example, in the early 1980s in the United States, the United States Football League attempted entry into the professional football market by hiring a relatively small number of star athletes. The league failed after only three seasons. The World Hockey Association’s entry into the professional hockey market in the 1970s met a similar fate in 1978. There do exist semipro teams and minor leagues in many major sports, but these leagues typically do not employ any of the top talent, and are often owned and operated by major league affiliates. The structure of the industry is determined in large part by the availability, cost, and distribution of the stars. Explaining the ‘‘market for stars’’ and the subsequent output market structure amounts to analyzing vertical product differentiation through competition for high-quality labor. The goal of this paper is to devise an entry model of professional sports markets incorporating this competition for stars (thereby endogenizing the cost of quality), and to discuss the implications on the market structures. Professional sports markets offer excludable public goods, where production costs (mainly labor costs) are fixed and the marginal cost of serving an additional consumer is virtually zero. Improving product quality amounts to increasing the number of stars in a league, and therefore increases the fixed costs of production. Stars may display synergy in that there is a certain appeal to seeing stars play against one another. Also, due to player heterogeneity, there is a relatively small number of star players available. If competing leagues wish to hire stars, they must engage in some form of auction to purchase their services. An appropriate model of this industry must therefore treat quality improvement as a fixed cost, describe the effect of potential synergies (or lack thereof) among stars, and allow leagues to bid against one another to acquire star players. A common trait among existing models of vertical differentiation is that firms tend to differentiate their products to ease price competition. How improvements in quality affect costs is an issue on which existing models differ. Gabszewicz and Thisse (1979), (1980) and Wauthy (1996), for example, assume no costs of improving quality. Mussa and Rosen (1978); Shaked and Sutton (1983); Gal-Or (1983), and Moorthy (1988) assume that quality improvements increase variable costs of production. Bonanno (1987); Shaked and Sutton (1987), and (Beath and Katsoulacos, 1991, p. 127), among others, assume that improvements in quality require increases in fixed costs. In cases where quality affects fixed costs, improvements in quality (real or perceived) arise from increased spending on research and development, production technology, or advertising. Each firm can independently increase its own quality without directly affecting the quality of its competitors’ products. As such, the marginal cost of improving quality is exogenously determined. The application of such models to professional sports markets may not be appropriate. Since stars are scarce, competing leagues do not have the freedom to independently choose product quality because one league’s choice to employ more stars may necessarily

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mean that a competitor employs fewer. Competition for stars determines the cost of hiring a star, and the marginal cost of improving quality is endogenous.1 Such competition creates a new dimension in which firms can strategically interact. Related research on the manipulation of competitors’ costs includes Salop and Scheffman (1983), where a firm efficient in advertising or R&D can advertise or invest heavily in R&D, thereby forcing competing firms to do the same and experience larger increases in costs. Also, the Pennington Case (Williamson (1968)) illustrates a situation where an industry-wide wage contract raises the costs of smaller-scale, labor-intensive competitors more than larger, capitalintensive firms. In the present analysis, increasing wages to stars influences not the production costs, but the costs of increasing product quality. Jones and Walsh (1987) study the effect of the formation of the World Hockey Association (WHA) in 1972 on player salaries in the National Hockey League (NHL) of the United States and Canada. The authors demonstrate that the formation of the WHA eliminated the monopsony power of the NHL, and player salaries moved upward and close to their marginal revenue products. Player salaries have remained high since the WHA folded in 1978, leaving in place an effective deterrent to entrants considering hiring those players. In another example, from 1977 to 1980 the Australian broadcasting executive Kerry Packer established World Series Cricket, a rival international cricket league. He negotiated with and hired some 35 of the world’s top cricketers in forming his new league, and in turn forced the established leagues to hire less talented players. Not only did this rival league introduce innovations such as night matches, colored clothing, and improved television coverage, but it also considerably increased the salaries of the leading cricketers. Cricketers since then (in particular those from the West Indies) have come to learn their worth, and salaries have remained competitive. Though there does seem to be evidence that leagues compete in quality, studies of entry deterrence in professional sports markets have not considered this strategy. Fort and Quirk (1995), for example, describe the expansion of current leagues into new markets to deter entry, but do not consider leagues with overlapping or identical markets. Rascher (1997) states that new leagues experience no known rivalries or team loyalty and thus are disadvantaged. The present paper attempts to analyze the importance of quality competition independent of these other factors. To formalize quality competition, this paper describes a bidding mechanism to compete for stars, and uses it in conjunction with the fixed cost / zero marginal cost

1

Dixit and Grossman (1986) analyze competition for inputs in a different context. In their model, the authors examine the desirability of export promotion policies where a number of industries compete for the services of scientists of inelastic supply. A difference is that scientists do not improve the quality of the product, but allow for increased production.

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and synergy properties to analyze market structure. Market equilibrium consists of either a monopoly or a duopoly depending on opportunity costs of players and the size of the market, but one league always employs all the stars. The market is likened to a ‘‘natural monopoly in both quantity and quality’’, where aggregate surplus is maximized if only one league exists, employs all the stars, and is price regulated to zero profits. Absent regulation, consumers prefer competition to the monopoly. Section 2 of the paper formulates the model from the market characteristics. Section 3 examines the first period situation of an incumbent monopolist facing no threat of entry, and the trivial result of an output monopolist and a monopsonist in labor is described. Section 4 analyzes the possibility of entry in the second period. The number of star contracts, wage offers, and Bertrand prices are determined, providing the subgame perfect Nash equilibrium outcome for the model. Section 5 discusses the results and extensions.

2. An overview of the model The method typically applied in analyzing quality and price competition is for competing firms to first choose quality and then choose Bertrand–Nash prices based on the resulting quality pairs. A subgame perfect Nash equilibrium in the sequential game specifies equilibrium quality / price pairs. In the present case, Bertrand–Nash prices are determined by product quality pairs, but quality pairs are determined by the number and size of wage offers to stars. The model for stars becomes one of determining the subgame perfect Nash equilibrium triplet of the number of offers / wages / price. This equilibrium is used to describe the possibility of entry, the equilibrium market structure, the distribution of stars, and market efficiency. To determine the effect of competitive bidding for stars on industry structure, other barriers to entry are ignored.

2.1. Production Several basic features of professional sports leagues are important in modeling production. First, each league produces a public excludable good, where any league can produce a unit of the product (a schedule of games), can sell access to a large number of consumers at negligible marginal cost, and can deny access to nonpayers. Second, the primary input is labor, where the output quality depends on the quality of the players. Third, there exists a relatively small number of star players and a large number of mediocre players. Lastly, consumers derive value from watching stars compete against one another instead of against mediocres. This last feature implies that star players may display synergy. If so, the marginal

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quality from a star player is nondecreasing as the total number of star players increases.2 In the model, the league is considered to be the firm. This assumption is supported by Grauer (1983) and Weistart (1984), who state that due to financial interdependence among the teams of a league and the cooperative nature of team sports, professional sports leagues are in effect single economic units.3 The marginal cost of serving consumers is assumed to be zero. Each league requires n players of any quality in production, and fewer players results in no production. There exist S identifiable star players of equal ability, and to incorporate the scarcity of stars, it is assumed that S < n.4 There exists a large number of mediocre players (called simply ‘‘mediocres’’). Any player’s utility is defined as V 5 w 2 r,

(1)

where w equals the wage earned in this industry and r equals the opportunity cost of accepting the job. The value of r is at least as great as the highest wage a player can receive in another industry, and may increase with any collective bargaining agreements. It represents the lowest acceptable wage.

2.2. Consumer utility Denote the number of potential consumers as Q. These consumers reside in the geographic locations in which the leagues operate franchises. Each consumer is fully informed of all characteristics of each product, and consumes either 0 or 1 unit of one product. The cost to the consumer is the price of a ticket to attend one game. Consumers are heterogeneous in preferences and income, and prefer

2 Star synergy seems to exist in professional sports, as evidenced by the USFL and the WHA. The few star players of each league had extremely strong personal statistics, but fans were not inclined to watch them because they were competing against less talented players. Rosen (1981) analyzes the convex relationship between talent and revenues of individual superstar performers. The superstar’s service being an excludable public good helps drive the results since a small number of sellers are able to serve a large number of buyers. Consumers can choose the superstar’s product over one of lesser quality without affecting the marginal cost. 3 Other assumptions regarding the economic unit of analysis may be as compelling and may affect the results. For example, since players typically contract with teams and not leagues, individual teams may be considered the firms and the league a coalition of firms. Still, teams compete for stars, and must still offer wages such that competing teams, inside or outside of the league, do not choose to hire them. Such a model may be used to determine the optimal distribution of stars among the teams of a league. Another view is that synergy results from adding teams with star players to the league. This analysis may be used to determine the optimal number of teams in a league, and in turn the number of stars employed. 4 Sports leagues often create expansion franchises (or merge with competing leagues) to accommodate an increased pool of talent. Allowing S .n can eliminate competitive bidding for the stars, and the model reverts to the standard one of independent quality choice. This possibility is therefore not explored.

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watching star players to watching mediocres. Due to player synergy, a consumer’s marginal utility is nondecreasing in the number of stars. For simplicity in the model, a standard linear utility function is assumed.5 As such, if a consumer with taste u and income Y pays p for a unit of a product produced with s stars, utility equals U 5 Y 2 p 1 u (a 1 b s),

(2)

where a, b .0. A positive a implies that a consumer derives utility from a product produced with no stars. The value u is assumed to be uniformly distributed on the interval [0, 1], and reflects the consumer’s marginal willingnessto-pay for quality. Player synergy is addressed in subsequent discussion demonstrating that the results are robust to convex forms of utility. Concave utility is also discussed, illustrating that the grouping phenomenon may be weakened if player synergy is not present.

2.3. The bidding mechanism and the timing There are two periods considered in the model. In the first period, there exists only an incumbent league and no threat of entry. A single potential entrant appears in period two. Entry implies production of the product, and for now requires no costs other than those associated with hiring players. Each period consists of three stages. In the first stage each star solicits offers by potential and existing leagues. Each league then simultaneously submits bids to a number of stars. A bid is an enforceable contract specifying wages for that period in return for labor services.6 Individual rationality states that no player accepts a wage that provides negative utility, so w must be greater than or equal to r. Incentive compatibility states that each player accepts the highest offer received. Due to any potential risks associated with a new league (e.g., likelihood of folding), it is assumed that if a player receives identical offers from the leagues, he

5

An interesting example concerns the Major League Baseball players strike of 1994–95, where the league threatened to use replacement, lower-quality players at lower salaries. The ticket prices were to be discounted, and in the case of the Texas Rangers, discounted ticket prices were to be determined by the formula S2A P 5 ]] (.5R) 1 .5R, B2A where A5value of all replacement-player salaries, B5value of all regular-player salaries (determined by the salary cap), S5actual sum of players’ salaries (determined by the salaries of replacement and regular players who do play), and R5regular ticket price. (Source: The Wall Street Journal, January 20, 1995) The ticket prices (and presumably willingness-to-pay) were to be an increasing, and on average linear function of the number of the regular star players employed by the team. 6 If a league does not produce, it is not obligated to pay its players.

S

D

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signs with the incumbent.7 It is further assumed that if any of a league’s bids are rejected, it may extend those bids to any remaining stars, if they exist. This last assumption guarantees that no star remains unemployed if a league wishes to hire him. Each league’s product quality is determined by its success in the star player ‘‘auction’’. In the second stage, any remaining vacant positions at each league are filled with mediocres. Since there is a large number of mediocres, all offers specify a wage r and are accepted. In the third stage production takes place, and each league announces the price for its good. Prices, like wages, are effective for the entire period. A subgame perfect Nash equilibrium characterizes the triplet ( f i* , w *i , p *i ) for each league i where f i* denotes the number of star offers made, w i* denotes the size of the star wage offers, and p i* denotes the Bertrand–Nash price. The combinations of f i* and w *i determine the equilibrium number of stars each league employs, s i* , which in turn determines prices. The period 1 solution is considered first, where there exists one incumbent facing no threat of entry. Following sections analyze the possibility of entry in period 2.

3. Period 1: The market with no threat of entry Other things equal, an incumbent monopolist facing no threat of entry can maximize demand for its product by employing all S stars and n2S mediocres. Since there is no competition for their labor in the first period, the incumbent hires each star at wage r. If the incumbent charges a price pI , denote u 9 as the lowest-type consumer who purchases its product. Demand for the monopolist’s product at price pI therefore equals (12u 9)Q. From (2), we see that the value u 9 is defined such that u 9(a 1 b S)5pI . Demand thus equals

S

D

pI qI 5 1 2 ]]] Q. a 1 bS

(3)

Since demand is linear in price and marginal cost is zero, the monopolist maximizes revenues and profits by pricing to serve half the market. Thus, q *I 5Q / 2, and the equilibrium price equals

a 1 bS p I* 5 ]]]. 2

7

(4)

In reality, players usually require a considerable premium to sign with an entrant league, making entry even more difficult. For example, in 1972 Bobby Hull required (and received) $1 million per year to sign with the Winnipeg Jets of the World Hockey Association. This amount was far greater than the typical star salaries of the time.

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Equilibrium revenues thus equal (a 1 b S)Q R *I 5 ]]], 4

(5)

and profits equal R *I minus the wage bill, nr.

4. Period 2: the market with potential entry In period 2, there exists an incumbent monopolist and a single potential entrant. The triplet ( f *, w*, p*) for the incumbent and the entrant is solved by first expressing equilibrium prices as functions of product qualities, because equilibrium revenues determine the most each league can pay any number of stars to achieve each quality pair. Thus, at the pricing stage, define s I as the number of stars employed by the incumbent, and s E as the number employed by the entrant (the remaining players are mediocres). Also, assume for the moment that s I .s E .

4.1. The demand functions It is assumed for simplicity that competing leagues draw from the same market, and therefore compete for the same consumers.8 A consumer of type uˆ is indifferent between the incumbent’s product at price pI and the entrant’s product at price if Y2pI 1 uˆ(a 1 b s I )5Y2pE 1 uˆ(a 1 b s E ). We can see that pI 2 pE uˆ 5 ]]]. b (s I 2 s E )

(6)

A consumer of type greater than uˆ purchases one unit of the incumbent’s product, and demand for the incumbent’s product thus equals (12 uˆ )Q. More formally, p 2p 1 2 ]]]DQ S q 5 5 b(s 2 s ) I

I

E

I

E

if pI < pE 1 b (s I 2 s E );

(7)

zero otherwise.

Demand for the entrant’s product equals ( uˆ 2u] )Q, where u] is the lowest-type 8

It is implicitly assumed that the entrant creates franchises in the same geographic locations as the incumbent. This assumption can be relaxed, where the entrant creates at least one franchise in locations not served by the incumbent. Demand for the entrant’s product may benefit, but demand for the incumbent’s product benefits as well. This assumption would not alter the competition for the stars, but would increase the profitability of entry with no stars. The reader should also note that the model assumes no team loyalty, and as such the incumbent cannot benefit from it. If team loyalty were to exist, demand for the incumbent’s product would become less elastic and the entrant’s more elastic. Entry would thus be less profitable and more difficult (see Rascher (1997)).

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consumer who purchases the entrant’s product at pE . The value u] is defined such that u] (a 1 b s E )5pE . Thus pE u] 5 ]]]. a 1 bsE

(8)

From uˆ and u], we see that demand for the entrant’s product equals p 2p p ]]] 2 ]]]DQ S q 5 5 b(s 2 s ) a 1 bs I

E

E

I

E

E

E

S

D

a 1 bsE if pE < pI ]]] ; a 1 bsI

(9)

zero otherwise.

If s I 5s E , each consumer chooses the least expensive product. The firm charging the lower price receives the full market demand while demand for the other firm’s product is zero. If prices are equal, the firms split demand.

4.2. The Bertrand–Nash equilibrium prices Demand being thus defined in the duopoly situation, the entrant considers the profitability of entry. Product quality is fixed, and since the marginal cost at this stage is zero, the goal of each league is to maximize revenues. If s I .s E , maximizing the incumbent’s revenues R I 5pI qI with respect to pI yields the incumbent’s price reaction function

b (s I 2 s E ) pE pI 5 ]]] 1 ]. 2 2

(10)

Maximizing the entrant’s revenues R E 5pE qE with respect to pE yields the entrant’s price reaction function

S

D

pI a 1 b s E pE 5 ] ]]] . 2 a 1 bsI

(11)

Solving (10) and (11) simultaneously yields the Bertrand equilibrium prices 2b (s I 2 s E )(a 1 b s I ) p *I 5 ]]]]]]; 3a 1 4b s I 2 b s E

(12)

b (s I 2 s E )(a 1 b s E ) p *E 5 ]]]]]]. 3a 1 4b s I 2 b s E

(13)

From demand, we see that equilibrium revenues equal 4b (s I 2 s E )(a 1 b s I )2 Q R *I 5 ]]]]]]] ; (3a 1 4b s I 2 b s E )2

(14)

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b (s I 2 s E )(a 1 b s I )(a 1 b s E )Q R E* 5 ]]]]]]]]] . (3a 1 4b s I 2 b s E )2

(15)

If s E .s I , equilibrium price and revenue values for the incumbent and the entrant are reversed. Note that as s E grows closer to s I , equilibrium price and revenue for each league approach zero. This is due to the fact that the leagues are offering more similar products, and can no longer rely on differences in consumer preferences to ease price competition. The leagues therefore compete more vigorously in prices. If s I 5s E , both leagues offer the same quality product. Pure Bertrand competition results, where a best response to a competitor’s price is to undercut that price by a small amount. Since leagues are acting to maximize revenues, prices are driven to zero (the marginal cost), and each league earns zero revenue in equilibrium.

4.3. The Nash equilibrium number of star contract offers The equilibrium revenues indicate each league’s maximum wage bill for any quality pair. The values s I and s E are determined by the number of star contract offers, fI and fE , and the relative sizes of the offers, w I and w E . The following proposition demonstrates the entrant’s equilibrium number of star offers in stage 1 of the bidding process: Proposition 1. If S
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Fig. 1. Equilibrium duopoly revenues and wage bill from hiring s stars.

distance between the revenue and wage bill represents profits. The proof shows further that profits are decreasing up to S / 2 due to the fact that revenues drop as products become less differentiated, and profits are increasing from S / 2 to S due to the fact that the marginal revenue from hiring a star exceeds the highest feasible marginal cost. Also, as the figure is drawn, profits are positive at s50. This need not be the case, as is outlined in the following section. The value nr may exceed revenues at s50.9 A second important result following from Proposition 1 is that since the entrant chooses f E* 5S, the incumbent also chooses f *I 5S. If it is profitable for the entrant to hire S stars and trade places with the incumbent, then it must be the case that the incumbent’s profits are greater than the entrant’s. The incumbent therefore is not willing to trade places, and chooses f I* 5S.

9 An informal illustration shows that the grouping result may rest in part on the distribution of consumer tastes because the density of consumers helps determine the marginal revenue from increasing quality. Suppose density equals f(u ), with cumulative density F(u ), where F(0)50 and F(1)51. If s I .s E , demand for the incumbent’s product equals (12F( uˆ ))Q, and for the entrant’s product equals (F( uˆ )2F(u ))Q. (Note that uˆ and u do not depend on the density.) If F(u ) is concave ] ] and if the incumbent increases quality, its quantity demanded and revenues increase at a faster rate than if F(u ) is linear. The conditions giving rise to the grouping result again exist (and are in fact stronger). However, if F(u ) is convex, the reverse is true for demand and revenues. Since revenues increase at a slower rate, the league may wish to stop short of hiring all the stars.

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4.4. The Nash equilibrium wage offers Given that each league makes offers to all the stars, the last step is to determine the size of the offers. In the ‘‘star auction’’, firms are assumed to follow a Bertrand-style bidding process. Each firm’s best response is to top (or match) its competitor’s offer as long as the resulting profits if the stars are successfully hired exceed those of the best alternative. In equilibrium, the wages are set such that at least one firm is rendered indifferent between hiring the stars and hiring none. If the entrant acquires no stars and decides to enter, then s I 5S and s E 50. Equilibrium prices can be found from (12) and (13), and from (14) and (15) we see that revenues equal 4b S(a 1 b S)2 Q R *I 5 ]]]]] ; (3a 1 4b S)2

(16)

ab S(a 1 b S)Q R *E 5 ]]]]] . (3a 1 4b S)2

(17)

If the entrant’s wage bill from paying each of the n mediocres a wage of r is less than or equal to R *E , such entry is profitable. This is true if r
4b S(a 1 b S) Q ab S(a 1 b S)Q ]]]]] 2 Sw *E 2 (n 2 S)r 5 ]]]]] 2 nr. (3a 1 4b S)2 (3a 1 4b S)2

(18)

Solving for w *E and simplifying yields

b (a 1 b S)Q w *E 5 r 1 ]]]]. (3a 1 4b S)

(19)

The incumbent in equilibrium chooses w I* 5w *E , and thus s *I 5S and s E* 50. This

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situation can be explained in Fig. 1. Competition for the stars increases the slope of the wage bill line to the point where the distance between the revenue and wage bill at s5S is equal to the distance at s50. The entrant is thus indifferent between those two outcomes, and enters with no stars. If r.r*, entry with no stars is not profitable. Therefore, the league losing the bidding war for stars does not hire mediocres and does not produce. The highest star wage offer the entrant would make is the zero profit wage if it becomes a monopoly. From monopoly revenues in (5), we see that the equilibrium wage is (a 1 b S)Q (n 2 S)r w *E 5 ]]] 2 ]]]. 4S S

(20)

If there are no sunk costs to entry, the incumbent again matches the entrant’s offer with w I* 5w *E , thereby successfully hiring the stars. Entry does not occur, the incumbent charges the monopoly price in (4), and earns zero profits. Entry is therefore only successful (with all mediocres) if r
10

Note that the bidding game is similar in nature to a contestable market in that the incumbent sacrifices its own profits (through high wages) in order to prevent entry. One may wonder to what extent this holds in practice, in particular if there is no established competing league to compete for the stars. In their analysis of the WHA’s entry into professional hockey in 1972, Jones and Walsh (1987) offer some justification to this assumption. They find that, as a result of direct competition for the players, salaries increased significantly after entry. However, after the WHA folded in 1978, the NHL was unable to reestablish the lower salaries despite its renewed monopsonistic power. This evidence suggests that the monopsonist league may have then kept salaries high since entry was proven to be a real threat. The authors portray the NHL’s subsequent attitude using a variation of an English rhyme: ‘‘WHA go away, don’t come back another day.’’ 11 Frascatore (1997) examines the strategic manipulation of multiperiod labor contracts to increase the cost to an entrant league of hiring stars.

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real threat of entry to salaries in a league with little threat but with free agency could serve to distinguish these effects.

5. Discussion Depending on the opportunity costs of players and market size, the market in equilibrium is either a duopoly producing maximally differentiated products or a monopoly producing the highest-quality product. Regardless of market structure, however, the incumbent employs all the stars. This grouping phenomenon is driven by two factors. The first is that hiring a star separates product qualities at twice the rate of unilateral quality changes, and the benefits to product differentiation are greater. The second is star synergy. Proposition 1 proves the grouping result assuming weak synergy in the form of linear utility, since the marginal revenue from hiring a star always exceeds the marginal cost if the league employs more than half the stars. (Marginal revenue is negative at less than half the stars.) It is easy to see that this grouping result also holds for convex utility. If utility is convex in quality as opposed to linear, then demand increases at a faster rate as quality increases, and revenues increase at a faster rate as well. The marginal revenue of quality again exceeds its marginal cost, and one league hires all the stars. Since star wages depend on the revenues from hiring all the stars and not on the shape of the revenue function, they increase only if utility from the highestquality product increases. This grouping result may indicate why we typically do not see oligopolies in professional sports where each league employs a portion of the best talent. Attempts at entry with a mix of stars and second-tier talent have, by and large, failed. The results of this model do suggest that more than one league is possible if the entrant offers a lower-quality game and charges a lower price. In the United States, the All-Star Football League recently considered entering the professional football market with second-tier players, while planning to compete in cities with and without NFL franchises. The results of this model suggest that this is the appropriate entry strategy. In Europe, the 1995 Bosman ruling significantly increased competition for star soccer players among Member States of the European Union by increasing the mobility of players at the end of their contracts. The ruling also eliminated limits on the number of citizens of other Member States that a club can field. Since the increased competition among the leagues, the European Federation, UEFA, and the leading European clubs have met to discuss forming a European club competition, with the long-run potential of developing into an elite independent European soccer league. Such a development, if ever realized, may provide further evidence of the tendency to group the top talent into one league. In examining market surplus, we see from the model that given zero marginal cost and positive fixed costs, aggregate surplus is maximized when each star is

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employed by one league at the reservation wage and the zero-profit price is charged.12 The league may be viewed as a ‘‘natural monopoly in quality’’ as well as in quantity. As such, it would be socially optimal to permit one league to employ all the stars and serve all the customers, and then to regulate the league’s price. Absent regulation, low-end entry is preferred to an unregulated monopolist. In the event of such entry, the model predicts that more consumers participate in the market, and more purchase the high-quality product. Aggregate surplus is thus improved. If there is entry, however, history indicates that if the entrant league does not fold, it often merges with or is partially absorbed by the incumbent league. This tendency towards monopolization certainly has not been resisted in the United States. For example, the federal government exempted the merger between the American Football League and the National Football League in the 1960s, and encouraged the National Basketball Association to merge with the American Basketball Association in the 1970s (see Fort and Quirk (1995)). The present model suggests that it may be in the interest of consumers for the federal government to encourage low-end entry and to resist subsequent merger activity. While the conclusions of the model suggest such policy, further analysis is necessary to determine the robustness of the results. The results rest in part on the assumption of star synergy. If utility is concave, the marginal revenue from hiring a star drops more quickly. It is possible then that the marginal revenue equals the marginal cost at some interior number of stars. In this case, the incumbent may offer contracts to a smaller number of stars, allowing the entrant to hire the remainder at the reservation wage if it wishes. Further entry with zero stars may then be possible. The absence of synergy and sunk costs of entry may give rise to the possibility of several tiers of product qualities in equilibrium, even with a small number of stars. Another limitation to the model is that the results are not entirely consistent with similar industries. For example, while films and city orchestras do tend to employ actors and musicians of similar talent, there is no one picture studio nor orchestra company that groups the best stars. Faculty of similar talent do tend to group at universities, but there certainly may exist more than two universities in equilibrium. Relaxing certain assumptions of the model or specifying new ones may yield results more consistent with these markets. For example, increasing the number of levels of worker quality may allow for the grouping of separate tiers of talent at different firms. Furthermore, if products are congestible and if there exist managerial diseconomies beyond some capacity, a single firm may not be able to effectively serve the entire market. The limit properties in such a setting would

12

Aggregate surplus is maximized with the constraint that the league earns at least zero profit. Marginal cost pricing actually produces a fully efficient market, but the league would exit the industry.

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change and perhaps allow for various product qualities. Further study could serve to identify the grouping phenomenon of workers under different contexts.

Acknowledgements Special thanks go to Simon Anderson, Irene Henriques, Jon Vilasuso, and three anonymous referees for valuable comments. I also thank two referees and participants at the Southern Economic Association Conference, New Orleans, LA for comments on an earlier paper that I have incorporated here. The usual disclaimer applies.

Appendix 1 Proof of Proposition 1. The entrant chooses to hire stars to the point where the marginal revenue from hiring an additional star equals the marginal cost of doing so, assuming the second order conditions hold. (To facilitate the proof, assume continuity in s.) The proof shows that if the entrant hires more than half the stars, the marginal revenue always exceeds the highest feasible marginal cost. If the entrant hires less than half the stars, the marginal revenue is negative. There therefore exist local profit maxima at zero and S stars. The entrant thus attempts to hire all the stars, and employs zero if it fails. Part 1 of the proof examines the case where S5n and the entrant hires more than half the stars. In part 2, S5n and the entrant hires less than half the stars. Part 3 expands the proof to cases where S ,n. Part 1: S5n, s E .S / 2 The marginal cost to the entrant of hiring a star instead of a mediocre is the wage offered by the incumbent minus r. From (14), we see that if the incumbent is to employ all the stars, it earns revenues equal to 4b S(a 1 b S)2 Q R *I 5 ]]]]] . (3a 1 4b S)2

(A1)

If S5n, the greatest wage the incumbent can pay each star thus equals 4b (a 1 b S)2 Q w 5 ]]]]] , (3a 1 4b S)2

(A2)

and the greatest possible marginal cost to the entrant of hiring a star equals 4b (a 1 b S)2 Q ]]]]] 2 r. (3a 1 4b S)2

(A3)

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If the entrant hires s E .S / 2 stars (and the incumbent hires S2s E ), from (14) we see that the entrant’s revenues equal 4b (2s E 2 S)(a 1 b s E )2 Q R *E 5 ]]]]]]] . (3a 1 5b s E 2 b S)2

(A4)

The marginal revenue from hiring an additional star then equals dR *E 8b (a 1 b s E )(3a 2 1 4ab s E 1 5b 2 s E2 1 ab S 2 3b 2 s E S 1 b 2 S 2 )Q ]] 5 ]]]]]]]]]]]]]]]]]] . ds E (3a 1 5b s E 2 b S)3 (A5) This value is defined and positive over the interval s E [(S / 2, S]. To demonstrate that the marginal revenue exceeds the marginal cost for any feasible value of r, I assume r50. Subtracting (A3) from (A5) (with r50) yields the marginal profit to the entrant from hiring an additional star: dPE 8b (a 1 b s E )(3a 2 1 4ab s E 1 5b 2 s 2E 1 ab S 2 3b 2 s E S 1 b 2 S 2 )Q ]] 5 ]]]]]]]]]]]]]]]]]] dSE (3a 1 5b s E 2 b S)3 4b (a 1 b S)2 Q 2 ]]]]] . (3a 1 4b S)2

(A6)

At the point s E 5S, this value, simplified, equals 3

2

2

2

3

3

4b (3a 1 6a b S 1 5ab S 1 2b S )Q ]]]]]]]]]]]. (3a 1 4b S)3

(A7)

This value is positive, so profits are increasing at the point s E 5S. Differentiating (A6) with respect to s E yields 2

2

2

d PE 2 8b (2a 1 b S) (6a 1 4b s E 1 b S)Q ]] , 2 2 5 ]]]]]]]]]]] d sE (23a 2 5b s E 1 b S)4

(A8)

which is negative. Since the profit function is concave and the first derivative is positive at s E 5S, it must be increasing and monotone on the interval s E [(S / 2, S]. The entrant thus prefers to hire S stars to s E [(S / 2, S) stars. Part 2: S5n, s E ,S / 2 If the entrant hires s E ,S /2 stars and the incumbent hires S2s E , from (15) we see that the entrant’s revenues equal

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b (S 2 2s E )(a 1 b S 2 b s E )(a 1 b s E )Q R E* 5 ]]]]]]]]]]] . (3a 1 4b S 2 5b s E )2

(A10)

The marginal revenue from hiring an additional star equals dR *E ]] 5 ds E b (26a 3 2 10a 2 b s E 1 18ab 2 s E2 2 10b 3 s E3 2 4a 2 b S 2 28ab 2 s E S 1 24b 3 s E2 S 2 5ab 2 S 2 2 19b 3 s E S 2 2 4b 3 S 3 )Q ]]]]]]]]]]]]]]]]]]]]]], (3a 1 4b S)3

(A11) which is defined over the interval and is negative at s E 50. The second derivative equals d 2 R 2E 2b 2 (215a 1 b s E 2 8b S)(2a 1 b S)2 Q ]] 5 ]]]]]]]]]]] , d 2 s 2E (3a 2 5b s E 1 4b S)4

(A12)

which is negative. Since the revenue function is decreasing at s E 50 and is concave, it must be decreasing in s E over the interval s E [[0, S / 2). Since the wage bill increases as s E increases, it must be the case that profits decrease in s E over this interval. The entrant thus prefers to employ no stars to employing s E [(0, S / 2). Part 3: S ,n If S ,n, note that the marginal revenues on the intervals s E [[0, S / 2), and s E [(S / 2, S] remain unchanged. Given the negative marginal revenue while hiring less than half the stars, there still exists a local profit maximum at zero stars. Also, if S ,n and the incumbent employs all the stars, it must pay n 2 S mediocres the wage r. Therefore, the highest wage the incumbent can afford to pay each star equals 4b (a 1 b S)2 Q (n 2 S)r w 5 ]]]]] 2 ]]]. S (3a 1 4b S)2

(A13)

This is lower than the highest wage if S5n (A2). Therefore, the marginal revenue on the interval s E [(S / 2, S] still exceeds the highest marginal cost of hiring a star, and there exists a second local profit maximum at S stars. The entrant therefore chooses to bid for each star, but bids an amount such that profits after hiring all the stars is at least as great as profits from hiring no stars. If unsuccessful, the entrant employs zero stars. Q.E.D.

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