The distribution of earnings and the rules of the game.

I. Introduction

Extensive research links the distribution of earnings to a variety of exogenous and endogenous factors: innate ability, motivation, acquired human capital, skill, occupation, race, sex, unionization, mobility, work-leisure preference, risk preference, luck, and so on. Overlooked in the analysis of income inequality is the institutional technology, or constitutional setting, or rules of the game in which income is earned. One of the authors [11] has examined the effect of differences in the degree of economic, legal (civil rights), and political freedom on income inequality across nations. He found that a substantial portion of the variance in income inequality was due to differences in the rights of various groups to compete for income streams. Much of constitutional history's evolution can be interpreted as investing previously unequal groups (peasants, blacks, women, the disabled, and so on) with equal rights to compete for income streams.

In general, individuals are endowed with, or acquire through investment and training, talent that can be utilized in productive activities. The realization of an individual's talent in an activity is his or her performance. The dimensions of talent are too numerous, and perhaps too mysterious, to be precisely quantified. But, it is convenient to think of talent as being composed of broad dimensions like intelligence, strength and coordination, and of further refinements of these dimensions (e.g., language, math, or reasoning ability). In this way, an individual's "talent" for a particular activity can be thought of as a weighted total of the dimensional attributes he or she possesses, where the weight in each dimension is determined by the demands of the activity.

Performance is the realization of talent plus a random component. Its expectation is an exact proxy for talent. Any discrepancy between talent and expected performance is the result of omitting relevant attributes of talent. For example, ignoring the intangible attributes of talent (determination, intensity, hustle) that contribute to one's success leads to the apparent observations of individuals that fail to perform at levels commensurate with their talent and individuals whose performance appears to surpass the measure of their talent. It must be said of the former that they are talented but lack something, say, concentration and of the latter that they make up for a lack of talent with something else, for example, effort. When the abilities to concentrate and to put forth effort are included as attributes of talent this discrepancy disappears and except for random occurrences (luck) performance is synonymous with talent.

In production activities the weights (rewards per unit) placed on the dimensions of talent are determined by derived demand, through production technology, the supplies of the attributes of talent, and the organization of the talent market (rules that compensate the various dimensions of talent). Where the rules of the talent market are the same for all, the distribution of income is a function of the distribution of performance weighted by labor effort, where labor effort refers to the number of performances an individual undertakes and is not considered an attribute of talent. This is most transparent in piecework activities, where performance can be measured as output per period and labor effort can be measured as the number of periods committed to piecework activities. Then, if one controls for labor effort, income is a scaler multiple (determined by the value of output) of performance, and the distribution of income and the distribution of performance are identical.

Within different talent markets, unique characteristics make for predictable differences in the relation between the distribution of talent and income. For example, when the payments for performance are rank ordered (that is, where the margin of performance between individuals determines their rank in the payment hierarchy but not the margin of compensation), income will be only rank correlated with performance and the distribution of income will (depending on the payment schedule) be more or less equal than the distribution of talent. In team production settings, where the assessment of individual performance is more difficult, or in instances where the consequence of the employment of labor effort and talent is uncertain, the distribution of income will tend to be more equal than in piecework settings. Finally, when there are rents (defined as returns not associated with the application of productive talent) to be distributed, income and its distribution may bear little resemblance to the underlying distribution of talent.

In this paper we examine the effects of the rules of the game on the distribution of player earnings in the sports economy. The analysis builds on Lazear and Rosen [6], by extending the rank order tournament model to team sports and by investigating the implications of the rules of the game (and changes in the rules) on the distribution of earnings in individual and team sports. We focus on the earnings of athletes for several reasons. First, the rules of the game in sports are known and in some sports have undergone substantial change. Second, the dimensions of athletic performance and the technology of production, although complex, lends itself to fairly unambiguous modelling. Third, evidence presented below suggests that the distribution of talent within a sport is similar across time and across samples so that observed differences in the distribution of earnings can be ascribed to differences in the rules of the game. Comparisons across sports is somewhat more problematic as comparisons of talent are not possible. George Stigler noted that in each academic discipline there were at any time only a few superstars. Rosen [7] makes the same observation about entertainment fields, including sports. Implicit in these observations is the notion that the distribution of talent, at least the upper tail, is similar across these occupations. For our purposes it is sufficient that differences in the distribution of talent across sports, if such differences exist, are not positively correlated with differences in the distribution of income that our analysis predict. Finally, data exist to test our propositions about the effect of the rules of the game on the distribution of earnings within and across sports. Several propositions concerning the effects of different compensation formulas, of team versus individual production, of uncertainty in the labor and product markets, of employer monopsony power, and of changes in technology are developed and tested. Many of the implications of these findings, mutatis mutandis, can be generalized to any production economy.

II. Individual Production

In productive activities, talent is a function of physical, intellectual and emotional attributes valued by the requisite technology. For our purpose we think of each attribute as quantifiable, with some, like height, exogenous (endowed), and others, like strength, endogenous (determined by investment). Let there be K attributes of talent denoted k. Each activity determines a set of weights that, when applied to the attributes of individual i, defines his or her talent as [t.sub.i] = [[Sigma].sub.k][b.sub.k][a.sub.ik], where the measures t, b and a are, respectively, talent, weight and attribute.

The realization of talent is an individual's average performance. Some factors beyond talent will influence performance (e.g., conditions of health, the environment, time of day, even how much sleep one got the night before). Thus, at any time the observation of individual i's performance, [P.sub.i], is a random variable with expectation, [t.sub.i], and a random component, [e.sub.i].(1)

[P.sub.i] = [t.sub.i] + [e.sub.i]. (1)

The [e.sub.i] are assumed to be independently and identically distributed (iid), with E(e) = 0.

We think of individual performance as a serially repeated and rewarded event. Among the individuals competing in professional sports the distribution of earnings is determined by the distribution of awards for performance, the distribution of talent (expected performance), and by work effort (the number of competitions undertaken). Even at the same wage, work effort is not equal among all, because people differ in their willingness to trade leisure for income. In what follows, expected earnings should be understood to mean expected earnings per event, a labor-effort-constant measure. Such a measure is appealing for comparisons of income distributions, because it compares potential earnings before the choice to purchase leisure. Observed earnings that measure potential earnings after the purchase of leisure treat the leisure choice negatively.

In a rank-order tournament the reward schedule is pre-determined and a competitor achieves the next higher reward by advancing one position in the rank of competitors. Consider random pairings of individuals in one-on-one, rank-order tournaments. Prior to the revelation of the pairings, each individual confronts the field of competitors. To compare the expected earnings of two individuals engaged in repeated rank-order contests, define [p.sub.12] as the probability that individual 1 will out-perform individual 2 in any given contest. That is,

[p.sub.12] [equivalent to] prob([P.sub.1] [greater than] [P.sub.2]) = prob([e.sub.2] - [e.sub.1] [less than] [t.sub.1] - [t.sub.2]) [equivalent to] G ([t.sub.1] - [t.sub.2]), (2)

where E([e.sub.2] - [e.sub.1]) = 0 and G is the cumulative density function (cdf) of [e.sub.2] - [e.sub.1]. Given a reward structure that pays [W.sub.1] to the winner and [W.sub.2] (less than [W.sub.1]) to the loser, the expected incomes are

E[Y.sub.1] = [P.sub.12][W.sub.1] + (1 - [P.sub.12])[W.sub.2] and E[Y.sub.2] = (1 - [p.sub.12])[W.sub.1] + [p.sub.12][W.sub.2]. (3)

Since some attributes of talent are given (endowed), the cost of incrementing expected performance, [C.sub.i]([t.sub.i]), differs among contestants. For convenience, and without sacrificing any of the results, let individual 1 have a lower marginal cost of incrementing talent, 0 [less than] [C[prime].sub.1] (t) [less than] [C[prime].sub.2](t). Individuals develop talent to maximize expected returns, E[R.sub.i] = E[Y.sub.i] - [C.sub.i]([t.sub.i]), which requires

[Delta]E[R.sub.1]/[Delta][t.sub.1] = ([W.sub.1] - [W.sub.2])g([t.sub.1] - [t.sub.2]) - [C[prime].sub.1]([t.sub.1]) = 0 and [Delta]E[R.sub.2]/[Delta][t.sub.2] = ([W.sub.1] - [W.sub.2])g([t.sub.1] - [t.sub.2]) - [C[prime].sub.2]([t.sub.2]) = 0, (4)

where g = G[prime] is the probability density function (pdf) of [e.sub.2] - [e.sub.1]. We make the Nash-Cournot assumption that each labor market participant makes investments in talent assuming the level of talent of the competitor is fixed. Under this assumption, if a stable Nash-Cournot equilibrium exists, like the one depicted in Figure 1, [Mathematical Expression Omitted] and, from equation (2), [p.sub.12] [greater than] 1/2.(2)

Equation (3) suggests that uncertain job performance plays a pivotal role in the distribution of income. When from one time to the next it is uncertain who will perform best, inequality in expected earnings is: 1) positively related to the difference in talent, 2) positively related to the inequality in relative rewards, but (3) less than the inequality in the structure of rewards.(3) A system that pays the highest paid worker 1000 times what it pays the lowest paid worker will have more inequality than a system in which the magnitude is only 100 times. But how much more inequality depends on the variance of performance that permits upsets. Equation 3) reveals that when rank-order contests permit upsets, inequality in the size distribution of income decreases as additional "noise" is added to performance and the magnitude of the decrease in inequality is greatest when the difference in talent is least.(4) When there is no variance in performance ([p.sub.12] = 1) the distribution of income exactly mirrors the distribution of rewards, i.e., E[Y.sub.1] = [W.sub.1] and E[Y.sub.2] = [W.sub.2]. When there is sufficient variance in performance to permit upsets, the distribution of earnings is more equal than the distribution of rewards.

When a series of contests determines the ultimate winner, more contests (observations) will reduce the (sample) variance of performance and the likelihood of an upset. In men's and women's professional tennis the playing rules and distribution of awards are the same with one notable exception, there is a greater reliance on five set matches on the men's circuit while the women play three set matches. One implication of the model is that in professional tennis the distribution of earnings on the men's circuit will be less equal than the distribution of earnings on the women's circuit.(5)

How much inequality is removed by increased variance depends on the difference in talent levels. In competitions where some individual is almost always a winner (loser), because the talent difference is so great, the equalizing effects of greater variance in performance will be slight. In sports an interesting extension of the play-against-the-field framework of competition is that of seeded match play. In seeded match play, past performances are used to rank performers for seedings in match play, where the highest ranked players play the lowest ranked players in the earliest rounds of a tournament. Matching the best and poorest players in our framework has the effect of increasing the difference in talents, [t.sub.1] - [t.sub.2], and reducing the likelihood of an upset (1 - [p.sub.12]).(6) A second implication of the model, then, is that, ceteris paribus, tennis with seeded match play will have a less equal distribution of earnings than golf where the contestants play against the entire field of competitors.

The market value, V, of a sports competition is a positive function of the talents of the competitors. According to equation (4) tournament sponsors can increase talent and, hence, the value of the competitions by increasing the difference in rewards, [W.sub.1] - [W.sub.2], and thereby increasing the marginal return to investments in talent. But, a sponsor's ability to raise [W.sub.1] or lower [W.sub.2] is constrained by tournament cost and by the opportunity cost of the competitors. Raising [W.sub.1] increases V but at the same time increases the cost of prizes. When perfect competition for talent exists, [W.sub.1] + [W.sub.2] = V. With monopsony power, [W.sub.1] + [W.sub.2] [less than] V. In particular, let [W.sub.1] + [W.sub.2] = [Delta]V, where 0 [less than] [Delta] [less than or equal to] 1 is a parameter which reflects the degree of competition in the demand for labor. Reducing [W.sub.2] increases V, by encouraging competitors to invest in talent, and, at the same time, lower cost, but individual 2's expected income must cover investment and opportunity costs. For opportunity cost [Mathematical Expression Omitted], this constraint implies [Mathematical Expression Omitted]. Competition and player opportunity cost constraints uniquely determine [W.sub.1] and [W.sub.2] as

[Mathematical Expression Omitted].

Equation sets (5) and (3) together reveal that inequality in the size distribution of income is positively related to the value of the activity and to the degree of competition in the labor market, and inversely related to opportunity costs.(7) In addition, the profit constraint, [W.sub.1] + [W.sub.2] = [Delta]V, implies that, ceteris paribus, effective minimum awards, [Mathematical Expression Omitted], come at the expense of the winner's prize, [W.sub.1], and decrease inequality in the size distribution of income.(8)

III. Team Production

Team members work together to create value but compete among themselves for positions on the team. The nature of these competitions is the same as the rank-order tournaments discussed in the last section; the "winner" is promoted and is more highly rewarded and the "loser," even if only slightly less talented than the winner, is relegated to a backup role and receives lower compensation [6]. Subject to team profit and player opportunity cost constraints, the team will establish an intra-team salary structure (described in the last section) that maximizes the incentive to invest in labor talent.(9) Team performance technology and conditions in the labor and output market determine the amount that is to be allocated to the prizes in these rank-order intra-team contests.

Define team j's performance, [[Rho].sub.j], as team expected performance, [[Tau].sub.j], a function of the talent of the N individual team members, plus a random component, [[Epsilon].sub.j] according to,

[[Rho].sub.j] = [[Tau].sub.j] + [[Epsilon].sub.j] for [[Tau].sub.j] = [Tau] ([t.sub.j1], [t.sub.j2],..., [t.sub.jN]), (6)

where [Delta][Tau]/[Delta][t.sub.jn] [greater than] 0 and [[Delta].sup.2][Tau]/[Delta][([t.sub.jn]).sup.2] [less than] 0 for n = 1,..., N. We assume the [[Epsilon].sub.j] are iid, with E([[Epsilon].sub.j]) = 0. In competitions between teams the probability that team j defeats opponent o is [[Pi].sub.jo] = prob ([[Epsilon].sub.o] - [[Epsilon].sub.j] [less than] [[Tau].sub.j] - [[Tau].sub.o]) [equivalent to] F ([[Tau].sub.j] - [[Tau].sub.o]), where F is the cdf of [[Epsilon].sub.o] - [[Epsilon].sub.j].

Some professional team sports (auto racing, doubles tennis and team track) are organized by sponsors who promote rank order tournaments with more or less open entry. Play against-the-field in rank-order team tournaments implies that team j will be randomly paired with opponent o in competition for winner's prize [[Omega].sub.1] and loser's prize [[Omega].sub.2] so that team j's expected revenue is

E[T.sub.j] = [[Pi].sub.jo][[Omega].sub.1] + (1 - [[Pi].sub.jo])[w.sub.2]. (7)

A more common form of organizing team sports in North America is the league format (football, baseball, basketball and hockey). Teams are affiliated with a league and the league is delegated some control in the operation of the teams. Leagues may do as little as establish a common set of rules of the game and schedule contests or go so far as to decide league membership, the distribution of revenue and the allocation of player talent. The delegation of these rights has profound implications for the distribution of income. In the league format, team revenues derive from two sources: shared revenues, [Mathematical Expression Omitted], that are allocated among the J teams without regard to performance (e.g., national media revenues) and team specific revenues [R.sub.j] (e.g., a portion of home ticket sales). Because shared revenues are not specific to any one team, we assume they are determined by league-wide talent levels, [Mathematical Expression Omitted]. What remains of value, after league rights are exercised, is distributed among the teams according to who created the value. Typically only home, and perhaps visiting teams, have the right to gate receipts and media revenues from the sale of game rights not taken by the league. In addition, there are other revenues, like stadium advertising and the sale of team logos, specifically identified with the team and not given to the league. We assume that team specific revenues are responsive to expected team performance, [R.sub.j]([[Pi].sub.jo]) [9; 10]. In the league format, with random scheduling, expected team revenue per contest is

[Mathematical Expression Omitted].

As in individual tournament play talent at each team position creates value for the team that it allocates among competitors for the position as rewards for performance and to encourage individuals to invest in the acquisition of talent. The value of talent at position n, [V.sub.n], is the interval of the marginal revenue product of talent employed at the position.

[Mathematical Expression Omitted],

where f = F[prime] is the pdf of [[Epsilon].sub.o] - [[Epsilon].sub.j].

[V.sub.n] is to be allocated, all or in part depending on the competition for talent, to the rank-order, intra-team competition for position n. As such, [V.sub.n] defines V in equation (5) and the greater is [V.sub.n] the greater is inequality in the size distribution of income.(10) The competitive parameter, [Delta], in equation (5) quantifies the degree of inter-team and inter-league competition for talent in team sports.

In each sports league the competition for talent ([Delta], in equation (5)) has varied widely over time as the levels of inter-team and inter-league competition have changed. To the extent that leagues erect barriers to inter-team competition for talent, like the player reserve system and the amateur draft, and to the extent that there are no alternative leagues, [Delta] may be quite low. In recent years there has been a significant liberalization of the rules of free agency in professional team sports. Veteran free agency has existed in baseball since 1976, in basketball since 1982 and in football beginning in 1993. Limited veteran free agency (with compensation between teams) was introduced in hockey in 1974. In addition there is increasing competition for talent in professional sports from over-seas leagues. Our third prediction is that equality in the distribution of income within a team sport has declined with the introduction of free agency and will continue to do so as long as inter-league competition continues to grow.

Across the team sports there are several differences predicted to influence the distribution of income that can be jointly tested. First, the degree of labor market competition differs significantly across leagues. In the 1990 and 1991 period extensive veteran free agency existed only in professional baseball and basketball and both leagues faced competition from overseas leagues. In professional football there was limited (Plan B) free agency for about 15 players on each roster and modest competition from the Canadian Football League. Of the four North American sports leagues, hockey had by far the greatest barriers to inter-team competition for players and there was virtually no other league competing at their salary level for player talent. Prediction 3 implies that the distribution of income will be less equal in baseball and basketball than in football and most equal in hockey.

Several issues are highlighted by equation (9). First, all else equal, greater inequality in the marginal products of the various positions, [Delta][Tau]/[Delta][t.sub.n], creates greater inequality in the size distribution of income.(11) To the extent that a rule change alters the legitimate means of production and the marginal products of inputs it will also alter the distribution of income. Second, rules for the distribution of revenues influence inequality in the distribution of player incomes. League rules divide the marginal revenue generated by additional talent between shared revenues, [Mathematical Expression Omitted], and team-specific revenues, [Sigma][R[prime].sub.j]f([center dot]). Sharing team-specific revenues creates a common pool problem. Consider a league mandated transfer, T, from team-specific marginal revenue, [Sigma][R[prime].sub.j]f([center dot]), to shared marginal revenue, [Mathematical Expression Omitted].Then from equation (9),

d[V.sub.n]/dT = 1/J - 1 [less than] 0. (10)

Free-riding on the creation of shared revenues reduces the value of talent at each position, [V.sub.n], by a factor of (J - 1)/J and, by prior reasoning, will reduce inequality in the size distribution of income.

While hockey has a 100-0 gate split, there is no national television contract and the value of the individual contests is not high relative to the other team sports. In professional football all television revenues are socialized, gate receipts are shared 60-40, winning franchises quickly fill their stadiums to capacity so that marginal attendance revenue is zero, and local sources of additional revenue are quite small. In basketball a large national broadcast contract is evenly shared, the gate split is 100-0 and teams have some limited ability to increase attendance and local media revenue. In baseball team-specific local broadcast revenues can be quite significant, sell-outs are not common and the gate split is approximately 85-15. Prediction 4 is that these differences contribute to greater inequality in the marginal revenue of winning and to greater inequality in the size distribution of income in professional baseball and basketball relative to professional football and hockey.

Because of the nature of team production, the performance of the individual is often not entirely separable from the performance of the team [1]. In this setting, team managers must monitor individual contributions to prevent shirking and the selfish pursuit of individual goals at the expense of the team's probability of winning (owner's profit). Management talent, not player talent, captures the return to the interactive effects of performance. Team talent functions fill the spectrum from separable production of the form [Tau] = [Sigma][t.sub.n] (relay races) to weak-link production of the form [Tau] = min{[t.sub.n]} (auto racing). When marginal products are fully separable, position values are attributed all of their marginal product, [V.sub.n]. When marginal products are not so easily separated, some fraction off the value of a position is allocated to management as a return for the scarce ability to put the right resource in the right position. Thus, if the difficulty of separating individual contributions increases there will be less value allocated to a player position and greater equality of income among team members (not including management).

Baseball is unique among the four team sports for which we have data in that an individual's performance as a hitter, pitcher or fielder is largely independent of their teammates so that performance is separable and easily measured. In any single instance a hitter's performance is dependent on the pitcher he faces and vice versa and such other influences as the score, the inning and whether there is a runner on base, but over the course of a season these inter-player dependencies vanish as players perform in a wide variety of situations.(12) This is not true in basketball, football, and hockey, where many players are active at once and the performance of one player affects the performance of others. For example, it is impossible in football to measure a running backs performance independent of the quality of the offense line.(13) Prediction 5 implies greater inequality in the size distribution of income in baseball than in the other team sports.

Individual sports and team sports differ in that the former permit variations in labor-effort (the number and level of contests that one enters) while the latter do not. The maximum amount of labor-effort that one puts forth is a choice variable in golf and tennis so that some part of the income inequality in these sports reflect differences in the number of tournaments entered. In team sports the maximum amount of labor-effort is limited by the team's schedule and all teams play the same number of games during the regular season. Some team members may participate in fewer games because of injury and there may be some additional reward for post-season play, but our data are salaries negotiated prior to the season and does not include post season pay. We assume a positively sloped supply curve of labor-effort so that the more talented players enter the most (or at least the most rewarding) tournaments. Our sixth prediction is that inequality in the distribution of earnings will be greater in the individual sports than in the team sports.

IV. Empirical Evidence

The Lorenz dominance test for our hypotheses requires comparing entire distributions of earnings and the data needed to calculate any one distribution is considerable. We have been able to assemble eight complete sets of individual annual earnings in the four major North American team sport leagues (baseball, basketball, football and hockey), two complete sets of annual winnings in individual sports (men's and women's golf) and two incomplete sets (limited to the highest money winners) of annual winnings in individual sports (men's and women's tennis).(14)

If each data set were complete and the underlying set of distributional rules were appropriately altered (each successive distribution introducing only one change) these twelve sets would permit us to make eleven independent test of our hypotheses. As it is we can only construct ten complete distributions. Evidence derived from the other two must be considered less reliable. In addition, we had no control over the underlying set of distributional rules. Fortunately, when there are several rule differences between sets they predict similar changes in distributional equality so that a number of our predictions can be tested jointly if not separately. We test six hypotheses derived from the theoretical discussion of the previous section and two additional hypotheses regarding the underlying distribution of talent in sports. In each case empirical evidence validates the theory although two tests rely on incomplete data sets and two other hypotheses cannot be separately tested. Table I presents a summary of the predictions of the model.

In each of the comparisons that follow we explicitly assume that any differences in the underlying distributions of talent across time and across the sports are small and random and, therefore, cannot explain the large and systematic differences in earnings inequality that form the basis of our empirical findings. Except in limited ways, that we explore below, this assertion cannot be tested because talent cannot be measured.(15) In fact, the observation of talent as it is applied to the sport, i.e., one's performance, is itself a function of the rules of the game that determine the weights to be placed on various attributes of talent. These "playing rules," as distinct from the labor market rules we explore here, can be manipulated by the owner/sponsors of the sporting contests, altering the weights placed on talent attributes and hence the distribution of talent. For example, Major League Baseball has reduced the strike zone, lowered the pitching mound and moved it back from the plate to reduce pitchers' dominance in the sport. Similarly, professional basketball introduced the three point play and the shot clock to reduce the premium on height and football has narrowed the goal posts, moved them back and introduced the two point conversion to reduce its reliance on place kickers. These are viewed as profit motivated rule changes. Scully [10] argues that since professional sports select from the upper tail of the distribution of talent, an excessive weight placed on any one attribute would increase the skewness of the distribution of talent in that sport and create scarce "super" talent with exceptional bargaining power. To the extent there is a profit-maximizing distribution of talent for team sports this kind of rule change would work to equate the underlying distributions of talent.

We envision two types of tests for equality of the distribution of talent. First, when performance can be unambiguously observed and measured it suffices as a proxy for talent. In the individual sports of tennis and golf performance is simply winning, however technical constraints (tennis cannot be organized as a play-against-the-field contest) mean that playing rules will alter the distribution of performance. In the team sports the observation of individual performance is seldom unambiguous. However, one narrowly defined observation comes to mind. Consider the [TABULAR DATA FOR TABLE I OMITTED] offensive role in baseball where performance is reasonably measured in total bases.(16) We hypothesize that the distribution of total bases is the same over different samples because the distribution of talent has not changed. A comparison of total bases in Major League Baseball in 1972 and in the National League, which did not implement the designated hitter rule, in 1989 (the years prior to our earnings data) yielded gini coefficients of .502 and .488 respectively and Lorenz distributions that were not significantly different.(17) Second, when the rules of the game and the distribution of prizes is the same an observed equality of the distribution of earnings would imply an underlying equality of the distribution of talent. In golf the labor market has been segmented into men's and women's markets with the same rules and formula for distributing prize money. Our seventh prediction about the distribution of income across sports is that the distribution of income will be the same in men's and women's golf because the underlying distribution of talent is the same. The results support this hypothesis.

Our predictions are summarized in Table I. Predictions 3, 4 and 5 jointly imply that across team sports the greatest inequality in the distribution of earnings should occur in baseball which has extensive veteran free-agency and some foreign competition, separable player performance, modest revenue sharing and valuable contests. Basketball will have a somewhat more equal distribution of income than baseball because there is a greater reliance on team production. In addition a league-imposed salary cap may suppress the income of superstars. In all other respects the labor market arrangements in baseball and basketball are alike.(18) The distribution of income among players in football will be more equal than in baseball and basketball, because of limited free-agency [TABULAR DATA FOR TABLE II OMITTED] and inter-league competition, highly interactive player performance effects, and extensive revenue sharing. In hockey the distribution of earnings should be more equal than in any other league sport. It has the most restrictive player reserve clause, no viable alternative leagues, highly interactive player performance effects, and generates significantly less revenue.

We first compare the distribution of earnings among the team sports. The estimated quintiles, standard errors, and Gini coefficients for the Lorenz curves appear in Tables II-IV. The test for differences in the Lorenz curves is the standard Lorenz dominance test [3]. The measure of the distribution-free standard errors for Lorenz ordinates is given by Beach and Davidson [2]. The relevant test statistic [Mathematical Expression Omitted] follows the Student maximum modulus distribution. Lorenz dominance occurs, if there is at least one positive and significant T value in the first four independent quintiles and no negative and significant T values. The critical value for T at the five percent level is 3.93. Two distributions are considered equivalent, if there are no significant values of T. They cannot be rank compared, if both positive and significant and negative and significant values of T occur.

Data on the distributions of income in team sports are presented in Table II. All of the distributions in team sports are significantly different from each other. As predicted, the transitive rankings for the recent salaries show the highest earnings inequality in baseball (1991, 1990), followed by basketball (1990), football (1990), and hockey (1990). Comparisons for recent and past distributions of salaries within the sports of baseball (1973, 1990, 1991), basketball (1967, 1990), and hockey (1978, 1990) also are significantly different. Veteran free agency in baseball and basketball has dramatically increased earnings inequality. The Gini coefficient rose in basketball from 0.28 (1967) to 0.44 (1990) and in baseball from 0.37 (1973) to 0.54 (1991). While the effect is not nearly as large in hockey (0.22 in 1978 to 0.28 in 1990), there has been a statistically significant increase in earnings inequality.

The distributions of earnings in the individual sports are given in Table III. In men's and women's tennis the earnings samples are truncated (the 150 highest earning women and men with earnings in excess of $44,000) and we do not know the total number of players in each association. As we do not know the extent to which low income participants are under-represented comparisons involving tennis are problematic. Only the data on earnings in men's and women's golf are a complete rendition of the membership in their respective associations.

As predicted, there is no clear rank ordering between men's and women's golf. The tails of their Lorenz distributions cross and their Gini coefficients nearly are identical (0.634 vs. 0.639). This is not surprising as the distributions of prize money are the same and there are no substantive differences in the rules governing the two associations. Using golf for comparison, the Lorenz dominance tests reveal that there is greater earnings inequality in individual sports than in all of the team sports. This likely is the result of the ability to vary labor effort in individual sports.

Table III. Individual Sports

                         Women's   Men's     Women's   Men's
                          Golf      Golf     Tennis    Tennis
                          1990      1990     1990(*)   1990(*)

1st Quintile Ordinate    0.0066    0.0028    0.0511    0.0606
(Standard Error)         (.0016)   (.0004)   (.0073)   (.0060)
2nd Quintile Ordinate    0.0412    0.0256    0.1203    0.1528
(Standard Error)         (.0065)   (.0045)   (.0169)   (.0151)
3rd Quintile Ordinate    0.1379    0.1265    0.2130    0.1542
(Standard Error)         (.0178)   (.0153)   (.0287)   (.0187)
4th Quintile Ordinate    0.3429    0.3582    0.3645     .03955
(Standard Error)         (.0321)   (.0210)   (.0450)   (.0324)
Number of Observations      198       311       150       222
Gini Coefficient         0.6392    0.6341    0.5550    0.4695

* Data were reported for the top money winners only. Caution must be
taken when interpreting these results.
Table IV. Individual Sports Top 150 Prize Winners

                         Women's   Men's     Women's   Men's
                          Golf     Golf      Tennis    Tennis
                          1990     1990       1990      1990

1st Quintile Ordinate    0.0291    0.0719    0.0511    0.0753
(Standard Error)         (.0034)   (.0047)   (.0073)   (.0064)
2nd Quintile Ordinate    0.0919    0.1765    0.1203    0.1753
(Standard Error)         (.0109)   (.0099)   (.0169)   (.0142)
3rd Quintile Ordinate    0.2100    0.3207    0.2130    0.3179
(Standard Error)         (.0212)   (.0151)   (.0287)   (.0241)
4th Quintile Ordinate    0.4133    0.5354    0.3645    0.5181
(Standard Error)         (.0346)   (.0183)   (.0450)   (.0358)
Number of Observations      150       150       150       150
Gini Coefficient         0.5419    0.3734    0.5550    0.3961

Table IV compares the earnings distribution of the top 150 prize winners in golf and tennis. These distributions cannot be rank ordered, because there is a significant crossing after the first ordinate. However, except for the first ordinate in the distribution of income in men's tennis, golf dominates that of tennis in all of the other ordinates. Obviously, since we do not know the unobserved earnings in the lower quintile, comparisons of the first quintile are spurious. The results offer weak evidence of the proposition that play-against-the-field yields greater equality in the size distribution of earnings than seeded match play.

V. Conclusions

We have extended the Lazear and Rosen [6] rank order tournament model to team sports and examined the effects of the rules of the game (compensation formulas, team v. individual production, play-against-the-field v. seeded match play, product and labor market uncertainty, monopsony power, and changes in production technology (e.g., playing rules)) on the distribution of earnings in team and individual sports. The distributions of player earnings varies tremendously in sports (range of Ginis from .22 to .64). The empirical evidence indicates that transitive rankings for player salaries follow the predicted theoretical effects of the rules of the game in sports.

The institutional technology or rule space is not time or spatially invariant. In our own economy, the distribution of labor earnings has been affected by, among many things, a transition from simple, separable production technologies (piece rate pay) to a more complex pay structure (team production, weekly wages). While the complexity of rule space change over time makes it impossible to untangle the effect on income inequality in the aggregate, inter-occupational comparisons and intra-occupational comparisons in different settings can be explored fruitfully. Perhaps, the rule space induced differences in earnings inequality in these dimensions will be as profound as in the sports economy.

Academia provides an interesting and familiar non-sports example. Casual observation suggests that the distribution of earnings among college and university professors at predominantly teaching institutions is more equal than the distribution of earnings among professors at research institutions. This is to be expected. First, teaching is a team production process in which labor effort is not easily varied while research is more akin to piece work with a high variance of work effort among academics. Second, it appears that there is greater competition for the skills of researchers than for those of teachers. Finally, there is greater ambiguity (higher variance) in the measure of teaching performance than in the measure of research performance. Of course, ambiguity in either measure can be reduced by increasing observations so it is not surprising if income inequality among academics from one's graduation class increases over time.

1. For this formulation we rely on Lazear and Rosen [6].

2. Stability requires that [C[double prime].sub.1] be sufficiently larger than [C[double prime].sub.2] at the equilibrium talent levels. For a large prize differential it is sufficient that [C[prime].sub.2] equal [C[prime].sub.1] plus a constant and that marginal cost increase at an increasing rate.

3. In keeping with the empirical tests in the later sections we say that distribution A has less inequality than distribution B if distribution A Lorenz dominates distribution B. There is an increase in inequality whenever there is a redistribution (or set of such redistributions) from a lower proportion of the ordinate measure to a higher proportion of the ordinate measure other proportions constant. Then the prior Lorenz distribution dominates the subsequent distribution between and including the lower and higher proportions and are identical elsewhere.

Here, we have assumed that 1/2 [less than] [p.sub.12] and if upsets are possible [p.sub.12] [less than] 1 so that (3) implies [W.sub.2] [less than] E[Y.sub.2] [less than] E[Y.sub.1] [less than] [W.sub.1], i.e., that inequality of prizes is greater than inequality of expected earnings. From (2) note that [Delta][p.sub.12]/[Delta]([t.sub.1] - [t.sub.2]) = g([t.sub.1] - [t.sub.2]) [greater than] 0 when [p.sub.12] [less than] 1 and from (3) [Delta]E[Y.sub.1]/[Delta][p.sub.12] = -[Delta]E[Y.sub.2][Delta][p.sub.12] = [W.sub.1] - [W.sub.2] [greater than] 0. Therefore, increases in talent differentials transfer income from the poorer to the wealthier. Similarly, increasing the reward differential by a transfer from a lower to a higher reward, [W.sub.1] + c and [W.sub.2] - c, creates a transfer between contestants according to [Delta]E[Y.sub.1]/[Delta]c = -[Delta]E[Y.sub.2]/[Delta]c = 2[p.sub.12] and increases income inequality.

4. Rothschild and Stiglitz [8] define additional noise as adding weight to the tails of the distribution of e and, hence, to the tails of the distribution of [e.sub.2] - [e.sub.1]. From equation (2) for given talents, [t.sub.1] and [t.sub.2], additional noise implies [p.sub.12] decreases and E[Y.sub.1] and E[Y.sub.2] become more equal. Because the distribution of [e.sub.2] - [e.sub.1] has a single mode at 0, the closer is [t.sub.1] - [t.sub.2] to 0 the greater will be the reduction in [p.sub.12] from additional noise.

5. This and other implications of the model that are derived below are summarized in Table I.

6. Laband [4] demonstrates this for an assumed distribution of talent and seedings and supports it with empirical evidence from sports.

7. From (3) and (5) [Delta]E[Y.sub.1]/[Delta]V = [Delta], [Delta]E[Y.sub.1]/[Delta][Delta] = V and [Delta]E[Y.sub.2]/[Delta]V = [Delta]E[Y.sub.2]/[Delta][Delta] = 0, so that increasing V or [Delta] increases the expected earnings of the better player only and, therefore, increase inequality in the size distribution of income. From (5) [Mathematical Expression Omitted] so that increases in opportunity costs decrease inequality in the distribution of prizes and, as we have already seen, in the size distribution of income.

8. Interestingly, [W.sub.2] need not be positive although [W.sub.1] must be. [W.sub.2] can be made negative by charging entry fees to the contestants. Because [W.sub.2] decreases as the total award, [Delta]V, increases and as the likelihood of an upset, 1 - [p.sub.12], increases, entry fees are most likely when the activity is most valuable, labor markets are very competitive, and talent differences are small.

9. In the Lazear [5] wage model with team production the wage structure is more equal than that which maximizes worker incentives to invest in talent when greater wage differences increase counter-productive predatory behavior by workers seeking promotion.

10. [V.sub.n] increases for a specific position when marginal product, [Delta][Tau]/[Delta][t.sub.n], increases. This increases positional inequality (fn. 9) and overall inequality (fn. 3). [V.sub.n] increases for all positions when marginal revenue, the remaining terms in (9), increases. This creates a set of positional inequality increases, and increases overall inequality (fn. 3).

11. By all else equal we mean specifically the opportunity cost of team members, [Mathematical Expression Omitted], and the cost of talent development, [C.sub.1] and [C.sub.2] (that produce [t.sup.*] in (4) and [p.sub.12] in (2)), for the players competing for position n. Then, from (5) increasing inequality in the marginal product of positions transfers expected income from the more talented player at some lesser paid position(s) to the more talented player at some higher paid position(s).

12. This is the reason baseball has attracted so much attention in the economics of sport literature. When inter-dependencies are important baseball has developed performance measures, like batting average with runners in scoring position or sacrifice flies, to capture this.

13. At the extremes of separable performance, perhaps, are auto racing - where team performance is often reduced to the poorest performance of the car (mechanic), the pit crew, or the driver - and relay races - where team performance is the sum of easily measured performance. Unfortunately, we have no individual income data in either sport and what team winnings we have gathered in auto racing is tainted by a wide variance in number of races entered.

14. Today most major sports report the earnings of participants. The sources of the 1990-91 salary data are from standard newspaper and league sources. Older salary data had to be meticulously gathered from newspaper and court records. The baseball salary data for 1973 is courtesy of Rodney Fort. There were 223 observations in the sample. An under-representation of rookies was suspected. The sample was compared with the 1990 population of players, based on the distribution of years in the majors. Seven rookie salaries were added to match the distributions. Thus, the sample size is 230 players. Roger Noll provided an economic honors paper at Stanford by his student Kim Akers. The 1967-68 sample of nine NBA clubs covered about three-quarters of the roster. Some observations for stars were missing, but mostly rookie player salaries were under-represented. Data was added from our own files for missing star players and rookie salaries were added to fill the rosters. A total of 28 observations were added to the available 84 from the Akers data. The Gini on the Akers data was .2474 compared to .2804 for the expanded sample. We believe our corrected population of the complete roster of the nine of twelve NBA clubs for the 1967-68 season is an accurate rendition. Data for the individual sports were provided by the various associations that represent the players.

15. With extensive data on attributes and performance one could construct measures of talent and reveal through hedonic estimation or principal components the weights placed on talent attributes by the playing rules of the game. The distribution of these measures could then be compared across sports. However, this data is not available to the authors. Extraordinary resources are devoted in sports to collect data on physical attributes (e.g., height, weight, speed, strength) of players in an attempt to predict future performance. Even with extensive data teams fait to accurately predict performance.

16. Total bases is defined as slugging average x at bats + walks + stolen bases.

17. The test for Lorenz dominance is discussed below.

18. Both leagues mandate minimum salaries but they do not differ appreciably between sports. The salary minimums for baseball and basketball in 1990 were 16.9% and 16.3% of mean salary, respectively.

References

1. Alchian, Armen A. and Harold Demsetz, "Production, Information Costs, and Economic Organization." American Economic Review, December 1972, 777-95.

2. Beach, Charles M. and Russell Davidson, "Distribution-Free Statistical Inference with Lorenz Curves and Income Shares." Review of Economic Studies, October 1983, 723-35.

3. Bishop, John A., John P. Formby and W. James Smith, "Lorenz Dominance and Welfare: Changes in the U.S. Distribution of Income." Review of Economics and Statistics, February 1991, 134-39.

4. Laband, David N. "How the Structure of Competition Influences Performance in Professional Sports: The Case of Tennis and Golf," in Sportometrics, edited by B. L. Goff and Robert D. Tollison. College Station, Texas: Texas A and M University Press, 1990, pp. 133-50.

5. Lazear, Edward P., "Pay Equity and Industrial Politics." Journal of Political Economy, June 1989, 561-80.

6. ----- and Sherwin Rosen, "Rank-Order Tournaments as Optimum Labor Contracts." Journal of Political Economy, October 1981, 841-64.

7. Rosen, Sherwin, "The Economics of Superstars." American Economic Review, December 1981, 845-54.

8. Rothschild, Michael and Joseph E. Stiglitz, "Increasing Risk: I. A. Definition." Journal of Economic Theory, 1970, 225-43.

9. Scully, Gerald W., "Pay and Performance in Major League Baseball." American Economic Review, December 1974, 915-30.

10. -----. The Business of Major League Baseball. Chicago: The University of Chicago Press, 1989.

11. -----. Constitutional Environments and Economic Growth. Princeton: Princeton University Press, 1992.