A general theory of professional sports leagues.

• Author: Vrooman, John

The evolution of free agency in professional sports leagues spans two decades from the 1972 Flood v. Kuhn decision in Major League Baseball (MLB) to the 1992 McNeil, et al. v. NFL decision in the National Football League (NFL).(1) In spite of the apparent relaxation of eligibility requirements for free agency, ancillary institutional arrangements have severely limited player mobility during these two decades in all leagues but MLB. As part of a "revolutionary partnership," National Basketball Association (NBA) players agreed to a league-wide payroll cap for team salaries in 1984 in exchange for a guaranteed 53 percent of NBA designated gross revenues (DGR),(2) The ostensible purpose of the NBA payroll or "salary" cap was to control spiraling salaries and generate competitive balance within the league, but since its inception, the payroll cap has served to limit player mobility under free agency, and its effect on competitive balance is subject to question.(3) Although free agency eligibility requirements have been lenient in the NFL, strict compensation rules have thwarted player movement until McNeil. In spite of free agency concessions by NFL owners following McNeil, movement of players among teams will continue to be limited under The Collective Bargaining Agreement of 1993, because of the concurrent imposition of a payroll cap at 64 percent of league-wide DGR beginning with the 1994 season.(4) During the second decade of the free agency era, the payroll cap has coevolved with free agency as its countervailing companion, and the payroll cap has simultaneously emerged as the major, unresolved issue in the collective bargaining of all professional sports leagues.(5)

The arguments for the payroll cap and strict compensation rules are the same as those traditionally made for the reserve clause. First, it is argued that player salaries should be controlled so that teams could recoup their investment in the development of talent. Second, it is maintained that talent will gravitate toward large market teams under unbridled free agency, and that constraints are necessary to defy gravity and maintain competitive balance within the league. In opposition to these arguments, conventional economic theory has held that the distribution of talent would be the same under free agency as it was under the reserve clause, and that competitive balance would not be affected by institutional change. This argument was originally made by Rottenberg when he surmised "that a market in which freedom is limited by a reserve rule such as that which now (1956) governs the baseball labor market distributes players among teams about as a free market would [25, 255]," and later embellished by Demsetz as application of the "Coase theorem" [4] when he asserted: "No matter who owns the right to sell the contract for the services of a baseball player, the distribution of players among teams will remain the same [10, 17]." The invariance proposition was formalized in the work of Quirk and El Hodiri [11; 23], and recent studies have reached a consensus that there has been no change in competitive balance due to free agency in either MLB or the NBA.(6)

Thus, while the legal and institutional foundations of major sports leagues have been co-evolving in the era of free agency, conventional economic theory has not. Daly observes that "Rottenberg's invariance proposition proved compelling to so many economists, some of whom viewed its logic to be so unassailable as to constitute a proof of its validity. Its grip on economists thinking persists to this day [6, 14]." The purpose of this paper is to reconsider the implications of conventional economic theory within the institutional configurations of professional sports leagues as they have evolved. This immediately involves the reduction of conventional theory to a special case of a more general theory of sports leagues.

1. A General Theory

   The economics of professional sports has been preoccupied with the dual proposition that a large market team will dominate a small market team, and that the competitive imbalance will be invariant under a variety of institutional constraints designed to alter it. Quirk and El Hodiri [11; 23] developed an elegant model of a sports league in which: "The evidence points to the conclusion that the rules structure of professional sports is relatively ineffective in balancing playing strengths, and that the imbalance is due to the differences in the drawing potentials of franchises [23, 58]," or more simply "big cities have winning teams and small cities have losing teams [23, 45]." The purpose of this section is to reconsider the implications of conventional theory for the contemporary world of professional sports, and to explore the possibility that large market dominance might be negated by other economic and institutional characteristics of professional sports leagues and the teams that comprise them.

   The implications of conventional economic theory can be seen through discussion of a model of a simplified professional sports league with two teams, each with the revenue and cost functions:

   [Mathematical Expression Omitted]

   where: [R.sub.i] is the total revenue for team i

   [R.sub.0] is exogenous revenue for both teams

   [Alpha] is the market size revenue effect

   [Beta] is the revenue elasticity of winning

   [p.sub.i] is the home market size for team i

   [Mathematical Expression Omitted]

   [C.sub.i] is the total cost for team i

   [C.sub.0] is exogenous cost for both teams

   [Gamma] is the market size cost externality

   [Delta] is the cost elasticity of winning

   [w.sub.i] is the winning percentage for team i

   and the zero-sum restrictions of the league require:

   [summation of] [w.sub.i] = n/2 = 1,000 where i=1 to n.

   The profit maximization conditions for each team yield the general competitive balance equilibrium:

   [w.sub.1]/[w.sub.2] = [[[p.sub.1]/[p.sub.2]].sup.([[Alpha]-[Gamma]])/([[Delta] - [Beta]]). (1)

   The customary assumptions of conventional theory can be summarized for the parameters in this model:(7)

   Alpha: Market size creates a revenue advantage that results in large market dominance on the playing field, i.e., [Alpha] [greater than] 0 for [p.sub.1] [Gamma] [p.sub.2].

   Beta: Marginal revenue is assumed to be a nonincreasing function of winning and all teams are assumed to have similar revenue functions, i.e., [Beta] [less than or equal to] 1.

   Gamma: The price per unit of talent is assumed to be the same for all teams, i.e., [Gamma] = 0.

   Delta: Marginal cost is assumed to be a nondecreasing function of winning and all teams are assumed to have similar cost functions, i.e., [Delta] [greater than or equal to] 1.

   The model bridges two separately incomplete treatments of competitive balance found in the recent literature: Quirk and Fort [24] and Scully [26]. Quirk and Fort (QF) base their rendition of Quirk's earlier work with El Hodiri (QE) on the assumption that "both teams will face the same market cost per unit of playing strength, and hence the same cost to increase the team's win/loss percentage (emphasis added) [24, 273]." This assumption can only be valid if there are no externalities of market size and if marginal costs are constant and the same for each team regardless of its winning percentage.(8) QF's model amounts to a special case of (1):

   [w.sub.1]/[w.sub.2] = [[[p.sub.1]/[p.sub.2]].sup.[Alpha]/(1-[Beta]) (2)

   where [Gamma] = 0 [Delta] = 1 by QF assumption. This competitive balance equilibrium is shown (as presented in QF) in Figure 1 for the large market Team 1 (left to right) and the small market Team 2 (right to left). Under the constant marginal cost assumption of QF, playing talent will move from Team 2 to Team 1 for any competitive balance less than .600/.400. Equal playing strengths between Team 1 at B and Team 2 at C create a disequilibrium condition because the potential gains to Team 1 outweigh the losses to Team 2 by the triangle ABC. All potential gains from talent redistribution for either team will be exhausted at A. The alpha factor revenue advantage enjoyed by Team 1 results in the league competitive balance equilibrium at A, and its .600/.400 dominance of Team 2.

   Following the invariance proposition, QF observe that A in Figure 1 will be the league equilibrium condition, regardless of whether the league is operating under free agency or such institutional player restrictions as the reserve clause or a salary cap. According to the invariance proposition, the economically justifiable distribution of talent in a Coasian world of zero transactions costs is independent of the legal ownership of that talent. The difference, of course, is that under free agency the players' salaries are commensurate with A, whereas under the reserve clause (or a salary cap), Team 2 will capture the gains and players will be paid less than the value of their marginal product at B. Conventional theory implies that the large market team will dominate the small market team with or without the institutional constraints on free agency, and that the difference under either regime relates not to the distribution of talent, but to the distribution of income derived from that talent.(9)

   Clearly the implications of (2) are that competitive balance is determined, not only by [p.sub.1] [greater than] [p.sub.2] and [Alpha] [greater than] 0, but also by the magnitude of [Beta], the revenue elasticity of winning. Solution (2) suggests the following beta factor elasticity propositions and attendant corollaries:

   PROPOSITION 1. If the revenue elasticities of winning are equal for the two teams, then competitive balance in the league will decrease and the large market team will become increasingly dominant as the revenue elasticity of winning increases. This is the move from A to B in Figure 2.

   COROLLARY. If the revenue elasticities of winning for each team are equal but decreasing, competitive balance will increase but, as long as market size is a significant factor, the league will...