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Bidding till bankrupt: destructive competition in professional team sports.

If we don't sign Hershiser, we'll lose him at the end of the year to an owner who is told by his manager that the club is only one pitcher away from winning it all. To win it all, some owners will do anything. God save us from those owners only one pitcher away.

Los Angeles Dodger owner Peter O'Malley (Durslag |1990~)


During the labor dispute that delayed the start of the 1990 baseball season, team owners raised their usual complaint that free agency will eventually result in a two-tier league of have and have-not teams, to the detriment of league revenues and fan interest. But player representatives remained skeptical of the validity of this complaint. Donald Fehr, executive director of the Major League Baseball Players Association, repeatedly pointed out that the owners have never demonstrated a need for "radical change" in the player compensation system, arguing, "They don't have a basis for it. We're looking for the rationale" (Newhan |1990, 10~).

This paper suggests such a rationale. In the pursuit of a league championship, the talent that turns an average team into a contender makes a disproportionately large contribution to the team's success. Owners therefore tend to earn most of their quasi-rents on the last stars they sign and suffer net losses on much of their infra-marginal talent. Teams with net losses on their overall stock of talent face a shutdown situation and will opt to abandon a competitive market for star athletes, reducing the number of remaining participants.(1)

It is useful to think of athletic talent in the sports industry as a "strategic input," an input whose productivity for a given user depends on how much of the input is employed by the user's rivals. The most well established point about inputs of this type is the tendency to overemploy them; in the sports literature this was demonstrated by Canes |1974~. This paper extends the analysis of strategic inputs in the professional team sports industry to demonstrate how the objective of championship prospects confronts market participants with a persistent shutdown threat. Of particular concern from an economics perspective, the shutdown scenario can result in a credible case of "destructive competition" in the sense of a competitive process that risks driving at least some participants out of a market even though it is inefficient for them to leave.

Sections II and III provide the theoretical basis for the shutdown situation which can emerge in the competition for athletic talent in professional team sports. Evidence for major league baseball presented in section IV suggests that the theoretical analysis may be empirically relevant. Section V closes with some implications of the analysis and possible extensions to situations outside the professional sports industry. The most important of the implications is the clear potential for welfare-reducing effects associated with the shutdown scenario.


Economists have long recognized that the market for athletic talent is not trouble-free. Researchers such as El Hodiri and Quirk |1971~, Demmert |1973~, Noll |1974~ and Daly and Moore |1981~ have discussed the inherent third-party externalities of player transactions involving any pair of teams because of the league-wide interdependence of relative team performance. Noll |1974, 416~ also noted that an unrestricted market for athletic talent can threaten the viability of marginal franchises who rely on rents derived from their players to cover some of their fixed costs. And Cassing and Douglas |1980~ modeled the winner's curse phenomenon in the context of an auction market for athletic talent.

But these analyses do not generate the bleak scenario described by team owners of a free-agency environment in which league divisions split sharply into have and have-not teams. For example, the third-party externality problem is not specific to free agency; it arises regardless of whether players or owners have the marketing rights to player contracts. Nor does it imply any sort of sharp dichotomy between successful and unsuccessful teams. Fixed costs can influence the entry/exit decision of each franchise, but they have no bearing on the allocation of variable inputs such as athletic talent among franchises that continue to survive in a free-agency environment. And the overbidding associated with the winner's curse is self-correcting in an ongoing competitive context.

But team owners may be right after all. The analysis which follows factors out previously established concerns about the market for star athletes and focuses strictly on athletic talent as a strategic input. Independently of any additional complications, the analysis yields a shutdown condition which can easily generate a sports league comprised of have and have-not teams and a suboptimal allocation of athletic talent.

The shutdown scenario rests on the productivity behavior of the athletic talent employed by a given team. The typical assumption about inputs is that diminishing marginal and average productivity set in early. Derived demand curves therefore tend to be downward sloping throughout most of the relevant range of input use. But athletic talent may not be so well-behaved. If fans tally the success of their preferred team in terms of perceived championship prospects then it is the last star acquired, not the first, that makes a disproportionately large contribution to team success. Increasing marginal and average productivity therefore persist well into the relevant operating region of a sports franchise.

The delayed onset of diminishing productivity turns on the pivotal role of championship prospects in motivating fan interest in the fan's preferred team. At least since Rottenberg |1956~, economists have acknowledged this role. But with few exceptions, such as Jennett |1984~ and Whitney |1988~, championship prospects have not been clearly differentiated from game-winning prospects in formal analysis. In their survey of the economics literature on professional sports, Cairns, Jennett and Sloane |1986, 11~ pointed out that economists usually measure team performance in terms of winning percentage. And the most well developed theoretical model of a sports league, by El Hodiri and Quirk |1971~, explicitly relates fan interest to victories rather than championships. The significance of the distinction is evident in the observation by former baseball player/manager Pete Rose that team owners "don't pay you for how many games over .500 you are. They pay you for winning" (Hersch |1987, 27~).

To highlight the distinction, consider a team sport league set up along the lines of the models laid out in El Hodiri and Quirk |1971~, Demmert |1973~, Canes |1974~ and Whitney |1988~. The league consists of m teams, with each team essentially an aggregation of athletic talent or playing skills (S). A total of |g.sub.T~ games per team constitutes a single playing season. Let the Indians (i) be one of the teams. Against rival team j the Indians play |g.sub.j~ games in a season, and their probability of winning each one (|P.sub.ij~) equals the Indians' share of the playing skills belonging to both teams:

(1) |P.sub.ij~ = |S.sub.i~/(|S.sub.i~ + |S.sub.j~).

Suppose first that fan support for the Indians depends on the team's expected winning percentage (|WPCT.sub.i~):

|Mathematical Expression Omitted~,

with the summation over j=1,m, j|not equal to~i.

The second derivative of |WPCT.sub.i~ with respect to |S.sub.i~ indicates that the marginal product of athletic talent in terms of expected winning percentage (|MP.sub.w~) is strictly decreasing:

|Mathematical Expression Omitted~,

with the summation over j=1,m, j|is not equal to~i.

In other words, when winning percentage considerations underlie the derived demand for athletic talent, input productivity is well-behaved: diminishing productivity sets in early, immediately in fact.

If fan interest depends on championship prospects instead of winning percentage, then success for the Indians depends not on a winning percentage that is high per se, but a winning percentage that is higher than those of the team's (m-1) rivals. Applying the voting paradox research of Garman and Kamien |1968~ and Niemi and Weisberg |1968~, the probability that the Indians will beat out their rivals to win a league championship flag (|PrF.sub.i~) depends on the simultaneous occurrence of (m-1) interdependent events:

|Mathematical Expression Omitted~


|Mathematical Expression Omitted~;

|Mathematical Expression Omitted~;

|Mathematical Expression Omitted~; and

|Mathematical Expression Omitted~.

The probability of each possible season outcome (|Mathematical Expression Omitted~) is in turn given by

|Mathematical Expression Omitted~,


|product~ = the product operator, indexed from j=1,m-1 and k=2,m; and

|v.sub.jk~ = a specified number of victories by team j over team k.

Niemi and Weisberg |1968, 319~ have pointed out that no closed-form expression for Pr|F.sub.i~ exists. However, the asymptotic behavior of Pr|F.sub.i~ emerges clearly from an examination of small-sample situations.

To eliminate the complications of third-party externalities, consider first a baseball league consisting of just two teams, the Indians and Yankees. The league employs a given total stock of athletic talent (the best available players); the Indians acquire some of the talent, and the Yankees employ the rest. Figure 1 illustrates the behavior of the marginal product of athletic talent for the Indians under these circumstances. The horizontal axis measures the amount of the league's talent employed by the Indians, with a unit of talent defined as one percent of the league's total stock of playing skills. The vertical axis measures the marginal product of each unit of talent in terms of its contribution to the Indians' championship probability.

The length of a playing season distinguishes the various marginal product curves in Figure 1, with the index in each case denoting the number of games played between the two teams. Each of the championship-related marginal product curves results from using equations (5) and (4) to compute the Indians' probability of a championship with progressively higher levels of talent and then subtracting the team's championship probability with talent lagged one unit.

Note in Figure 1 how, in contrast to the continuously diminishing marginal product of athletic talent in terms of winning percentage, marginal product in terms of championship probability increases over a sizable range of athletic talent in multigame seasons. In fact, diminishing marginal productivity doesn't set in until the Indians become the stronger of the two teams. Of course, diminishing average productivity sets in at an even higher level of employed talent. Note also that the more games the Indians play, the higher and sharper the spike in marginal productivity.

These results make intuitive sense. On any given day, a team with any skill at all can beat its rival. That's the popular interpretation of a Bernoulli trial. But over the course of a playing season, the law of large numbers progressively pushes weak teams out of a championship race and strong teams into it. And neither the very weak nor the very strong gain much from acquiring a little more talent, particularly when the season is a long one. The most productive star is the one that gives a team a competitive edge in the race for its league championship.

Outcome distributions for sports leagues with additional teams reveal a similar pattern for the productivity of athletic talent in terms of championship probability. When a team faces equally talented rivals, its marginal productivity peaks in the neighborhood of the team's pro-rata share of league talent. When a team faces unequally talented rivals, diminishing marginal productivity sets in later, when the team's level of talent is in the neighborhood of the skill level of the league's leading championship contender. Since diminishing average productivity lags diminishing marginal productivity, at least some teams, particularly the weaker teams in a league, are therefore forced to operate in the region of increasing average productivity of athletic talent.(2)


With the delayed onset of diminishing productivity, the viability of an interior optimum for teams competing in the market for star players becomes an open question. Ordinarily, with the early onset of diminishing marginal and average productivity, firms collect positive quasi-rents on most of their infra-marginal units of an input. But in what Neale |1964~ labelled the "peculiar" business of professional team sports, the conventional productivity sequence gets reversed in the market for star players: sports teams are likely to incur losses on a sizable portion of their infra-marginal athletic talent, losses which may not be offset by positive quasi-rents on talent nearer the margin. The possibility therefore arises that one or more teams will face a shutdown situation in which it becomes more profitable to retreat to the sidelines of a competitive market for athletic talent than to remain at an interior optimum.

The key consideration is the behavior of marginal revenue product, which depends on both the extra output that an additional unit of an input produces and on the revenue that the extra output fetches. A sports team can experience diminishing returns in the face of increasing productivity, but only if fan revenue exhibits a sufficiently weak response to increased team success.

To see this, suppose that each team j in a sports league collects revenue only from the willingness-to-pay of its home-market fans and incurs expenses associated only with player and talent acquisition costs. Assume that each fan appreciates simply having a team to root for as well as the preferred team's championship prospects and therefore contributes revenue according to the function

(6) R = |R.sub.o~ + Pr||F.sub.j~.sup.|eta~~.

With positive and diminishing marginal willingness-to-pay for improved team performance, |eta~, the elasticity of performance-related fan revenue (R-|R.sub.o~) with respect to championship probability, ranges in value between zero and one. Let each team consist of n players. Assume that if team j does not participate in the auction market for athletic talent, it ends up with a base talent level of |S.sub.o~ and incurs a cost per player of w, the common opportunity cost of the player's next best employment opportunity outside the sport. With |A.sub.j~ fans in the home market of team j, the team's profits are given by

|Mathematical Expression Omitted~

where C(|Mathematical Expression Omitted~) is the cost to team j of any transactions carried out to adjust the team's talent from its base level. The two bracketed expressions in (7) correspond to team j's "base profits" and its "performance profits," respectively. To abstract from conventional entry/exit concerns, let base profits be nonnegative for each team, so the existence of each franchise is both profitable and welfare-improving. For now, assume also that |S.sub.o~ equals zero.

In a competitive market for star players, each team should acquire talent until the marginal revenue product of the talent equals its unit cost. But participation in the market is profitable only if marginal revenue product (MRP) does not exceed average revenue product (ARP) at the interior optimum. This inequality establishes the following operating criterion for each team:

|Mathematical Expression Omitted~

Cancelling and rearranging terms yields

(9) |eta~ |is less than or equal to~ (Pr|F.sub.j~/|S.sub.j~)/(|delta~Pr|F.sub.j~/|delta~|S.sub.j~) |equivalent to~ (|AP/MP).sub.j~,

where AP refers to the average product and MP to the marginal product of athletic talent. For a team operating in the region of increasing average productivity MP exceeds AP; equation (9) suggests that the team still operates with diminishing average returns only if the elasticity of performance-related revenue with respect to championship probability is low enough to offset the gap between marginal and average product.

As an example, consider again a two-team league consisting of the Indians and Yankees. Figure 2 illustrates the performance-related revenue situation for a twenty-five game season in which fan elasticity equals 0.2 and market sizes differ in such a way that marginal revenue products are equated across teams when the probability of a championship equals 20 percent for the Indians and 80 percent for the Yankees. In the figure the marginal and average revenue products of athletic talent are measured from the left-hand origin for the Indians and from the right-hand origin for the Yankees. Note that at talent prices low enough to keep the Indians in the market, there is an excess demand for talent. But at the market-clearing price, the interior solution in Figure 2 is nonsustainable for the Indians.(3) The Indians generate performance-related revenues that are insufficient to cover the costs of their athletic talent, and the team can reduce its losses by shutting down.(4)

The shutdown threat does not necessarily stop with the weaker teams of a league, and it can in fact extend to all teams. For example, consider a league in which teams in equal-sized markets start from a position of complete parity. With a very long season, the slightest increment to the stock of athletic talent for one of the teams will be decisive. The leading team will win the league championship for sure, the other teams will lose for sure. Since the marginal star in this case makes the difference between complete success and complete failure, the market-clearing wage under competitive bidding equals a team's entire performance-related revenues. None of the teams can pay that wage to all of its players; all of them face a shutdown situation.


A quantum leap separates the theoretically possible from the empirically relevant. An overlapping mix of motivations appears to drive the interest of fans in their preferred team, and a variety of factors influences the actual performance of the market for star athletes in each sport.(5) These complications preclude a thoroughgoing assessment of shutdown prospects in professional team sports. The more modest goal of this section is to consider the case of major league baseball and offer some preliminary evidence which suggests that the theoretical case for the shutdown scenario laid out in sections II and III may in fact be empirically relevant to the professional team sports industry.

Table I reports some schematic information concerning the participation of the twenty-six major league baseball teams in the market for athletic talent during the years 1982-1989, the eight years following the baseball players strike of 1981. The results are mixed and difficult to interpret. For example, note from the correlation coefficients reported at the bottom of Table I that there is no statistically significant positive association between team winning percentage and the number of free agents acquired from other major league teams during the sample period. However, teams can accumulate talent in other ways as well, such as player trades, amateur drafts and the retention of potential free agents. Since player salaries reflect these options as well as the impact of free-agent acquisitions, salary differences across teams should correspond more closely to relative aggressiveness in acquiring athletic talent. The data in Table I suggest that this is so: the correlation coefficient of 0.637 between team winning percentage and team salary relative to league average salary is statistically significant at a confidence level above 99 percent. And some of the least successful teams--the Texas Rangers, Seattle Mariners and Cleveland Indians--have average salaries considerably below the major TABULAR DATA OMITTED league average. However, it is not necessary to resort to the model of this paper to account for a positive association between team performance and average salaries, and it is not certain that any of the low-paying teams went so far as to shut down in the 1980s.

One of the most significant factors complicating the past performance of major league baseball is the fact that the league's free agency option has been restricted to players with six or more seasons of major league experience. Team owners retain the contract marketing rights for players with less experience, so the shutdown threat under a more thoroughgoing system of free agency remains an open question.

To anticipate the possible consequences of a freely competitive market for athletic talent, consider again the operating criterion specified in equation (9),

(10) |eta~ |is less than or equal to~ AP/MP.

Examination of the operating criterion indicates that several teams in each major league division, particularly the least successful teams, may in fact face a shutdown situation with unrestricted free-agency.

Table II records a variety of relevant productivity information for major league baseball teams, again covering the period 1982-89. The average winning percentage of each team during the period appears in the first column. Column 2 records the implicit and mutually consistent talent levels necessary to generate the observed winning percentage data, according to the formula given in equation (2):

|Mathematical Expression Omitted~,

with the summation over j=1,m, j|is not equal to~i.

I plugged the skill levels listed in column 2 into formula (1) |S/(|S.sub.i~ + |S.sub.j~)~ to generate bilateral victory probabilities and then estimated divisional championship frequencies by running a large number of simulated baseball seasons. Column 3 reports the estimated initial championship probability of each team and column 4 the estimated championship probability the team would experience with a small increase in its stock of athletic talent.(6)

Estimated average product, marginal product and the ratio of average-to-marginal product of athletic talent for each team appear in columns 5, 6 and 7 of Table II. The values of all the ratios in column 7 are below 1.0, which indicates that no major league baseball team is sufficiently dominant that it operates in the region of diminishing average productivity. For each team, the ratio in column 7 corresponds to the right-hand side of equation (10). Its value indicates how low the elasticity of performance-related revenue with respect to championship probability must TABULAR DATA OMITTED be in order for the given team to avoid a shutdown situation with unrestricted competitive bidding for athletic talent.(7)

Averaged over the four divisions, the threshold elasticities range from 0.042 for the weakest divisional contender to 0.185 for the strongest contender. To interpret these, consider a team that raises its average championship probability from 1.2 percent to 36.4 percent, thereby improving its performance from the average bottom team in a division to the average top team. The team's performance-related revenue would rise 15.4 percent with an elasticity of 0.042 and 88.0 percent with an elasticity of 0.185. In terms of total team revenues, these threshold values must be revised downward to reflect the inclusion of receipts unrelated to team performance. Given the low threshold elasticities at the bottom end of this estimated range, it certainly seems plausible that performance-related revenue could be sufficiently elastic to threaten at least the weakest baseball franchises with a shutdown condition if they are forced to acquire all of their talent at competitive prices.

An empirical estimate of the elasticity of performance-related revenue with respect to championship prospects can push this plausibility assessment somewhat farther. To compute this estimate, I used team home attendance as a proxy for performance-related revenue and assumed that, while spot attendance varies with spot performance, average attendance over time depends on a team's underlying championship prospects. I therefore employed a cross-section equation for the twenty-six major league teams to regress the log of average home attendance for the years 1982-89 (ATT) on the log of estimated championship probability (PrF), the logs of population (POP), per capita income (YCAP), ticket price (P), and stadium capacity (SEATS), and two dummy variables, one to tag teams facing competition from another baseball team in their home market (BB) and the other to tag American League teams (AL). The regression analysis yielded the following estimated coefficients, with standard errors reported in parentheses:(8)

|Mathematical Expression Omitted~.

The 0.168 estimate for the elasticity of attendance with respect to championship probability lies above the estimated upper-bound operating elasticities in Table II for all but three of the major league baseball teams. And the least successful teams in their respective divisions--in the American League, Cleveland in the East and Seattle in the West, and in the National League, Pittsburgh in the East and Atlanta in the West--all have estimated upper-bound elasticities roughly four or more standard deviations below the estimated coefficient on PrF. This additional evidence suggests again that competitive bidding for star athletes may in fact push at least baseball's weakest teams past the point of shutdown.


A model in which championship prospects drive fan interest affords a number of unique insights into the performance of the professional team sports industry. Most significant is the possibility that unrestricted competition in the market for athletic talent will result in destructive competition and, consequently, suboptimal social welfare. To see this, observe in Figure 2 that under unrestricted free agency, the Indians abandon the market for athletic talent even though it is inefficient for them to do so: comparing the heights of the marginal revenue product curves, the talent released by the Indians generates more revenue for them than for the Yankees.

Moreover, the allocation of athletic talent becomes sensitive to the type of contracting arrangement implemented by the league. Economists such as Rottenberg |1956~, El Hodiri and Quirk |1971~, Demsetz |1972~, Demmert |1973~, Noll |1974~ and Holohan |1978~ have argued that the incentive to sell athletic talent to the team which values it most highly assures that the allocation of talent among teams will be the same under free agency as under other contracting arrangements (such as baseball's former reserve clause) which grant team owners the marketing rights to player contracts. Here, the prospect of shutdown in combination with the increasing costs that rival teams impose on each other trips up this argument. Under a reserve clause, a team in a situation similar to the Indians in Figure 2 could profitably attract the talent it values more highly than its rivals by offering to purchase player contracts from rival team owners for prices that exceed the revenue the rival owners receive from employing the talent themselves. Successive transactions elicit lower bid prices and higher ask prices, but each transaction remains mutually beneficial until the interior optimum is reached. Arrangements like the reserve clause essentially accommodate a type of price discrimination that is unworkable when contract marketing rights belong to the players themselves. Alternatives to free agency can therefore result in an allocation of players that is different from, and more efficient than, the allocation that results when players market their own services.

Similarly, a player draft plus tenure requirement prior to eligibility for free agency can also help teams avoid a shutdown situation. A player draft increases the base level of talent (talent level |S.sub.o~ in equation 7) that a team can achieve without bidding for free agents. The likelihood of subsequent participation in the free agent market rises since the team enters the market somewhere to the right of the origin in Figure 2 and faces a reduced range of purchased talent over which the team incurs losses. Teams which still confront a shutdown situation can at least operate closer to the interior optimum with the talent they acquire via the draft itself. This interpretation resolves the apparent inconsistency in the asymmetric behavior of small-market teams which avoid bidding for free agents but simultaneously refuse to part with the talent they acquire from outside the free-agent market. As observed by Daly and Moore |1981~, Cymrot |1983~ and Dolan and Schmidt |1985~, player drafts and reserve clauses may in fact influence the allocation of athletic talent among teams, possibly for the better. Salary arbitration, on the other hand, works in the opposite direction, increasing the prospect of shutdown by reducing the portion of talent which a team can acquire at bargain prices.

More even sharing of gate receipts among teams offers another route for reducing the threat of shutdown. Quirk and El Hodiri |1974, 67~ demonstrated that more even gate sharing among teams improves franchise profitability by reducing each team's marginal revenue product of athletic talent and therefore the equilibrium level of player salaries. But in their model gate sharing does not influence the allocation of talent across teams. The latter conclusion changes here. By increasing the profitability of smaller franchises at their interior optimum, a sufficiently high rate of gate sharing can rescue them from the shutdown condition.(9) More even gate sharing therefore constitutes a possible response to free agency that can alter the allocation of athletic talent in a way that improves efficiency.

Perhaps the most intriguing implication concerns just how accurate Neale |1964~ was to label professional team sports a "peculiar" industry. In the analysis here the market for athletic talent possesses two key distinguishing characteristics: (1) a delayed onset of diminishing returns, which raises the odds of a shutdown; and (2) a situation of increasing costs, driven by the revenue losses teams impose on each other, which introduces the possibility that the prospective shutdown is inefficient. Strategic inputs are certainly not unique to the professional team sports industry: consider, for example, research skills in patent races or combative advertising in product promotion. The open question is whether strategic inputs elsewhere exhibit characteristics similar to athletic talent in professional team sports, with similar implications for market concentration and social welfare.

A dynamic model of the professional team sports industry or a long-run model similar to that of El Hodiri and Quirk |1971~ might lead to results different from those derived here, though it is not apparent that either extension would undo the key conclusions of this paper.(10) Two characteristics of the analysis, however, do distinguish the results here from those that might emerge under different conditions. The first is the assumed irrelevance to productivity of absolute levels of employment of the strategic input, an assumption which might be debatable even in professional team sports. The second is the presence of demand asymmetries facing individual producers; such asymmetries are clearly absent when consumers have no reason to prefer a particular rival on the basis of location or identity. Further research might indicate the consequences of relaxing these constraints.

Cairns, Jennett and Sloan |1986, 3~ noted that the sports industry is a productive testing ground for a variety of topics of professional concern, such as antitrust, cartels, racial discrimination and monopsonistic exploitation. For these purposes, the industry typically serves as an example, not as an exception. It remains to be seen whether the same ultimately proves true in the case of destructive competition.

JAMES D. WHITNEY Associate Professor, Occidental College. I wish to thank Ed Earl for his research efforts, Manuel Pastor for his computer software assistance, and Stephen Bell, Susan Feigenbaum, Shane Greenstein, James Lehman, Robert L. Moore, two anonymous referees and economics seminar participants at Occidental College, the Claremont Graduate School and the 1990 WEA International Conference for their valuable comments on earlier drafts of this paper.

1. In microeconomic theory, shutdown receives perfunctory treatment. Its empirical prospects are considered remote, except as the first step toward exit. It lacks a well-specified enabling mechanism within the production process. And efficiency tends to rise when it does in fact occur. The analysis here challenges all three of these characteristics of the conventional shutdown scenario.

Related corner solution possibilities (for individual work effort) appear in the research on optimal labor contracts by Lazear and Rosen |1981~, Nalebuff and Stiglitz |1983~ and O'Keeffe, Viscusi and Zeckhauser |1984~. However, such corner solutions are more endogenous at the interfirm level of internal labor markets than at the interfirm level of industrial organization. The emphasis in this paper accordingly shifts from the determination of the optimal structure of an economic contest to an examination of the outcome of a rivalrous situation in which the "contest" itself is exogenous.

2. This result is quite pronounced and robust. With the perfectly inelastic aggregate supply of talent assumed here, all teams in a competitively balanced league operate in the region of increasing average productivity since diminishing marginal productivity does not set in until after a team acquires its pro-rata share of talent. When teams acquire their additional talent from outside the league instead of from other teams, diminishing marginal productivity sets in earlier, but outcome distributions generated under this assumption indicate a negligible difference in terms of the likelihood that teams will operate in the region of increasing average productivity. With competitive balance, all teams operate with increasing average productivity if teams play each other more than five times in a two-team league, more than one time in a three-team league and one or more times in a league with more than three teams.

3. El Hodiri and Quirk |1971, 1307~ consider the uncertain existence and nature of an equilibrium allocation of talent when a sports league consists of a small number of teams. It is already well established that inefficiency typically accompanies interaction strategies that do not mimic competitive behavior. The point here is that strategies conducive to a competitive outcome also risk leading to an inefficient end result. Besides, it is not obvious that the twenty-six major league baseball teams or even the twelve National League baseball teams constitute small numbers for analytical purposes.

4. The fact that one team ends up monopolizing the league's talent is a peculiarity of the two-team case described here. With several teams, one or more teams might face a shutdown situation while one or more others remain active competitors in the market for athletic talent.

5. Among other complications, overlapping motivations for fan interest raise the possibility of multiple equilibria in the market for athletic talent.

6. I chose a skill level increment equal to 0.4 percent of each league's initial stock of talent, an increment that amounts to a 5 percent increase in the talent level of an average team. I made no offsetting reductions in the talent levels of rival teams, essentially assuming that the additional talent comes from outside the league. I used a sample of 12,000 simulated, regulation-length playing seasons to estimate the championship frequency of each team with its enhanced talent. The combination of sample size and skill level increment yielded sufficient confidence that the random samples would all move in the expected direction when examining marginal productivity. The larger samples used to construct the estimates in column 3 consist of a baseline set of 12,000 seasons, plus the samples generated whenever the team with enhanced talent did not belong to the listed team's division. (Small skill increments for teams in a listed team's opposing division did not appear to have a material effect on championship outcomes within the team's own division.)

Note that I have continued to assume that team talent is a linear function of individual playing skills. The actual production function is an open question, but it is interesting to note in this respect that Scully |1974; 1989~ and Medoff |1976~, in their empirical research, estimate team winning percentage as a direct linear function of player productivity. All else equal, their specification is even more conducive to increasing marginal productivity in terms of championship prospects than is the El Hodiri-Quirk specification adopted here.

7. In considering each team, this implicitly assumes that the team had not already been pushed to the point of shutdown by the limited free agency of the 1980s. For a team already opting to shut down, the breakeven elasticity would tend to be higher than the estimate reported in Table II, but the shutdown itself would constitute additional potential evidence of the empirical relevance of the model suggested here.

8. Data sources: Annual issues of Statistical Abstract of the United States, World Almanac and Book of Facts, The Official Baseball Dope Book, The Sporting News, Scully |1989~, and the 1985 edition of Sales and Marketing Management's Annual Survey of Buying Power. As in Whitney |1988, 708~ and mostly attributable to Demmert |1973, 58~, I combined the following SMSAs: Los Angeles-Santa Ana, San Francisco-San Jose, New York-Newark, Baltimore-Washington, D.C., and Dallas-Fort Worth. I used 1984 cross-section data for the slow-moving variables of population, income, ticket price and stadium capacity (stadium capacity deviated significantly from its 1984 level only for the Toronto Blue Jays and for them only in 1989, when the team moved to a new stadium).

Acknowledging that teams generally operate as local monopolies and plausibly maximize gate receipts (assuming a negligible marginal cost of additional attendance), a more appropriate specification entails regressing gate receipts instead of attendance on the variables included here except ticket price. Assuming stable relative ticket prices, multiplying average attendance by 1984 ticket prices to estimate gate receipts and carrying out the revised regression yields virtually the same estimated coefficient on PrF, (0.167 instead of 0.168).

9. To see this, refer again to Figure 2, and imagine a two-team league with team 1 measured from the left-hand origin and located in a smaller market than team 2, measured from the right-hand origin. The curves illustrate the marginal and average revenue products (MRP and ARP) of athletic talent for the two teams in the absence of gate sharing.

Quirk and El Hodiri |1974, 67~ have proven that a stable and unchanged interior equilibrium allocation of athletic talent (S*) exists as long as the home team's share of gate receipts (|alpha~) exceeds one-half. As |alpha~ falls with gate sharing, MRP and therefore the equilibrium price of athletic talent fall as well. For team 1,

MRP1 = |alpha~(|delta~R1/|delta~S1) + (1-|alpha~)(|delta~R2/|alpha~S1).

At S*, (|delta~R2/|delta~S1) = - (|delta~R1/|delta~S1), so MRP1

at S* simplifies to MRP1 = (2|alpha~-1)(|delta~R1/|delta~ S1).

MRP1 falls with |alpha~ and approaches zero as |alpha~ approaches its lower limit of 0.5.

ARP1, however, remains positive as |alpha~ approaches 0.5:

|Mathematical Expression Omitted~.

Note in Figure 2 that as team 1's level of talent rises from 0 toward the interior optimum, the size of the revenue gain to team 1 exceeds the size of the revenue loss to team 2 (|delta~R1/|delta~S1 |is greater than~ -|delta~R2/|delta~S1). Since |alpha~ also exceeds (1-|alpha~), the value of the integral above is strictly positive. Since ARP1 remains positive while MRP1 approaches zero as the rate of gate sharing approaches its upper limit of 0.5, MRP1 must eventually drop below ARP1 at S*.

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