Spillovers, complementarities, and sorting in labor markets with an application to professional sports.

I. Introduction

A moral of Aesop reads, "The strong and the weak cannot keep company." Casual empiricism indicates that law firms, hospitals, academic university departments, and many other industries seem to consist of companies with relatively homogeneous quality workers and that top firms are generally averse to hiring low-quality workers, even at significantly lower wages. One explanation for this behavior is the presence of labor complementarities caused by spillovers across workers, as when the presence of better coworkers improves a worker's own productivity.

Spillovers by themselves do not necessarily affect the sorting of workers into firms. However, spillovers can affect worker-firm sorting if the size of the spillover depends not only on the characteristics of the individual worker but also on those of his coworkers. In this case, the wage a worker can demand for his labor may differ across firms because the composition of workers in those firms differs. In particular, if higher-quality workers receive greater spillover benefits from high-quality colleagues, it is optimal for homogeneous quality workers to be matched together in firms as complementary labor inputs.

This paper provides theory to this effect and considers empirically the specific case of basketball teams. To illustrate the previous discussion, when a player with high offensive talent is added to the team, he tends to draw double- and triple-teaming defenses to himself, opening up clearer paths to the basket for his teammates and improving their likelihood of scoring. This constitutes a spillover. If the improvement in the likelihood of scoring the teammates receive is higher for high-quality teammates than for low-quality teammates, this constitutes a complementarity between players and suggests that the optimal matching of players to teams will tend toward homogeneous teams, with the best players matched together, since this is where their marginal physical product is highest.

I find that there are significant spillovers to offensive production in professional basketball: A 10% increase in teammates' productivity leads to a 4.5% increase in own productivity. However, there is little evidence for offensive complementarities: Teams are not significantly homogeneous in offensive talent.

This paper makes two primary contributions to literature. First, it provides a general equilibrium structural model linking productivity spillovers to worker sorting. Sorting models like the one developed here are part of the lengthy literature on "assignment" problems, in which a number of heterogeneous units are to be assigned into groups in such a way as to optimize the aggregate output. Von Neumann (1953) and Koopmans and Beckman (1957) present mathematical derivations of basic assignment problems. Becker (1973) provides a canonical model of coordination and assignment in marriages. Applying this model to the labor market, Kremer (1993) and Saint-Paul (2001) consider assignment of workers to firms when there are complementarities in production. The present paper extends this literature to specifically account for assignment when spillovers to productivity exist and to provide empirically estimable specifications.

The paper also adds to the literature on the estimation of production processes in sports. Scully (1974) presented estimates of baseball players' marginal revenue products. (1) Scott, Long, and Somppi (1985) and Berri (1999) use similar methods in basketball The accuracy of these estimates, however, depends on the absence of complementarities between players since productivity depends not only on the player but also on the teammates he played with, as Scully (1974) notes. Idson and Kahane (2000) find some evidence of labor complementarities in the National Hockey League, and Carmichael and Thomas (1995) provide evidence on complementarities between labor and other inputs in rugby. Chapman and Southwick (1991) find that baseball managers have different productivity levels when matched with different teams. This paper takes a different approach to the problem than previous studies; it seeks not to measure complementarities directly but instead to measure spillovers to offensive productivity between professional basketball players and then to seek appropriate evidence of matching to check for the existence of complementarities. Unlike previous studies, I utilize a panel data set to distinguish the existence of spillovers from any sorting effects they may generate. Moreover, I utilize the fact that offensive production in basketball is sometimes performed in teams (floor shooting) and sometimes individually (free throw shooting) to distinguish spillover effects from other exogenous effects on labor productivity. (2)

Section 2 develops a model of spillovers in a generalized workplace and shows the conditions under which these effects generate the complementarities between workers that lead to homogeneous matching of workers into firms. Section 3 explores professional basketball as a case study. Section 4 concludes.

2. The Model

The workforce consists of a unitary measure of workers. Workers can be classified as one of N > 1 types, indexed in the set Q = {1, 2, ..., N}. A worker of type j [member of] Q is of quality [[phi].sub.j], which might measure an exogenous endowment of human capital, for instance. There is a finite measure [m.sub.j] of type j workers, where [[summation of].sub.j[member of]Q] [m.sub.j] = 1.

Firms consist of coalitions of workers, and a particular firm k consists of an (endogenous) measure [n.sub.jk] of type j workers, so that the total measure of workers at firm k is [n.sub.k] = [[summation of].sub.j[member of]Q] [n.sub.jk].

All workers provide an inelastic quantity of labor; however, since the workers differ in quality, the input of quality-adjusted labor hours differs by worker type. Let the quality-adjusted labor input ("productivity") of a worker of type j at firm k be [a.sub.jk].

To quantify the spillover effects across coworkers, let [a.sub.jk] be a function of j's own quality, [[phi].sub.j], the total productivity of the other workers at firm k, and the size of the measure of other workers at firm k:

(1) [a.sub.jk] = g([[phi].sub.j], [A.sub.k], [n.sub.k]) where [A.sub.k] = [summation over (j[member of]Q)][n.sub.jk][a.sub.jk].

If [a.sub.jk] is increasing (decreasing) in the second argument, then there are positive (negative) spillovers. If larger teams are more difficult to coordinate than smaller teams, then g should be decreasing in [n.sub.k]. This means that if an additional worker is not supplying enough productivity to the coalition through his quality, he may be a net drag on team output because there are costs to including him in the production process. An example of this might be a case where a group of inventors are working on a new drug in a pharmaceutical company. Adding additional inventors increases the probability that someone will come up with a patentable idea, but in the process of developing such an idea, the inventors may argue among themselves about the best way to proceed in research or may find it difficult to come to a unified decision regarding production in a reasonable amount of time.

The output of the firm is [A.sub.k]:

(2) [A.sub.k] = [summation over (j[member of]Q)][n.sub.jk][a.sub.jk].

Equations 1 and 2 jointly describe the production process of firms in the economy. Together, these equations imply a production function for the firm:

(3) [A.sub.k] = [summation over (j[member of]Q)][n.sub.jk]g([[phi].sub.j], [A.sub.k], [n.sub.k]).

Define an "assignment" for a firm like that described in Equation 3 to be a set of numbers [n.sub.jk] for every j and every k such that

(i) [n.sub.jk] [greater than or equal to] 0 for all j and all k and

(ii) [[summation of].sub.k] [n.sub.jk] [less than or equal to] [m.sub.j].

An assignment both defines the number of firms in the economy and indicates which workers work at which firms. The first condition indicates that nonnegative assignments must be made of each type to each firm. The second condition indicates that the total number of type j workers assigned must not be more than the number of type j workers who exist in the labor force.

Now define an "equilibrium" in this economy as

(i) a wage function [w.sub.jk],

(ii) a positive number K indicating the number of firms, and

(iii) a set of numbers {[n.sub.jk]} for all j [member of] Q and for all k [less than or equal to] K, satisfying the following conditions:

(4) {[n.sub.jk]} is an assignment

(5) [summation over (j[member of]Q)][n.sub.jk][w.sub.jk] [less than or equal to] [A.sub.k] [for all] k [less than or equal to] K

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equation 5 indicates every firm can cover its labor costs. Equation 6 indicates that the assignment {[n.sub.jk]} is individually rational in the sense that no subcoalition of workers at any firm or any coalition of workers between several firms could produce more together than they are receiving in equilibrium. Equation 7 implies no unemployment.

The existence, uniqueness, and character of equilibria will generally depend on the functional form of g. Intuitively, though, there are two important conditions in determining the equilibrium. First, if one type of workers improves the productivity of their coworkers more than other types, then this value will be priced into their wages in equilibrium. Second, if types value the presence of like types more than other types, for example if the value of a high quality coworker is increasing in a worker's own quality, then each type of worker will bid their own type of coworkers away from the other firms. In this case, workers will sort together by quality, with equal quality workers tending to work together in the same firms. Alternatively, if a particular type's presence is valued equally by all workers or if his presence is valued more by other types than by his own, firms will be heterogeneous.

To formalize these ideas, it will be helpful to consider a special case of the function g:

(8) g([[phi].sub.j],[A.sub.k],[n.sub.k]) = [[phi].sub.j] [([A.sub.k]/[n.sub.k]).sup.[epsilon]].

Because of the multiplicative form, if [epsilon] > 0, the value of a high-quality coworker increases in the worker's own quality. The following proposition shows that this implies the existence of an equilibrium with homogeneous quality firms:

PROFOSITION 1. Let [epsilon] > 0. Then

[w.sub.jk] = {[[phi].sup.1/1-[epsilon]] if j = k 0 otherwise}

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an equilibrium for the function g specified in Equation 8.

PROOF. To see that Equation 6 holds, consider a firm that has a group of heterogeneous workers {[n.sup.*.sub.ik]}. The marginal product of a worker of type j in such a firm is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, for a given [epsilon], it can be checked that 1 - [epsilon] is the maximum of the function

[([phi]/[[phi].sub.j]).sup.[epsilon]/1-[epsilon]] - [epsilon][([phi]/[[phi].sub.j]).sup.1/1-[epsilon]],

and this maximum is attained when ([[phi].sub.j] = [phi]. Hence,

[([phi]/[[phi].sub.j]).sup.[epsilon]/1-[epsilon]] - [epsilon][([phi]/[[phi].sub.j]).sup.1/1-[epsilon]] [less than or equal to] 1 = [epsilon],

which implies

[[phi].sub.j]/1-[epsilon][([bar][phi]).sup.[epsilon]/1-[epsilon]] - [epsilon]/1 - [epsilon]([bar][phi])1/1 - [epsilon][less than or equal to] [[phi].sup.1/1-[epsilon].sub.j].

The left-hand side is the marginal product of a worker of type j in a heterogeneous firm, as derived previously, while the right-hand side is the worker's wage in equilibrium. Since this relationship holds with strict inequality for any [[phi].sub.j] [not equal to] [phi], any heterogeneous firm will be unstable in equilibrium. This proves that the equilibrium satisfies Equation 6. It is trivial to check that the equilibrium satisfies Equations 4, 5, and 7. QED.

In this equilibrium, a firm with many high-quality workers could hire some low-quality workers, but because they would have to deduct the cost of integrating them into the team from their wages, they would not be willing to offer a wage higher than the low-quality workers could have received at a firm with other low-quality workers.

Now consider a second functional form for g, namely,

(9) g([[phi].sub.j],[A.sub.k],[n.sub.k]) = [[phi].sub.j] + [epsilon]([A.sub.k]/[n.sub.k]).

The difference between the model with Equation 9 and the model with Equation 8 is that in Equation 9, the benefit of higher-quality coworkers is independent of a worker's own quality. In this case, any assignment of workers to firms can be supported as an equilibrium:

PROPOSITION 2. Let [epsilon] > 0. Then

[w.sub.jk] = 1/1-[[epsilon].sub.j] for all k, K = N, {[n.sub.jk]} = [m.sub.j]/N for all k

is an equilibrium for the function g specified in Equation 9.

PROOF. Equation 6 holds because each worker's marginal product is independent of the composition of his coworkers. Hence, any alternate assignment {[s.sub.jk]} would give wages exactly equal to those in equilibrium, and there would be no incentive for workers to leave the equilibrium. Again, Equations 4, 5, and 7 are trivial to check. QED.

Propositions 1 and 2 confirm the intuition given previously. If a particular type values like types more than other types (as in the model when g is multiplicative), then the wages of those types will be adjusted so that they sort together in the same firms homogeneously. If a particular type values other type coworkers the same or above his own type, then wages will adjust so that workers sort heterogeneously.

The remainder of this paper considers the model empirically in the case of professional sports in an attempt to see if spillovers exist and whether the form they take resembles more closely that of Equation 8 or Equation 9.

3. Estimating Peer Effects and Assignments

Methodology

One approach to estimating spillovers is to simply run a cross-sectional regression of individual productivities on team productivities. An example of such an approach might simply take the log-linear form of Equation 8:

(10) log([a.sub.jk]) = [alpha] + [epsilon]log([A.sub.k]/[n.sub.k]) + [[eta].sub.jk],

where [[eta].sub.jk] is an individually varying error term and the regression estimates the parameters [alpha] and [epsilon], where [epsilon] is the same parameter so denoted in the previous model.

I will discuss two general confounding empirical factors in the estimation of the parameter [epsilon] through this method and how they are to be ameliorated in the work that follows.

First, regressions of Equation 10 cannot distinguish productivity spillovers from the equilibrium impacts they have as long as data are taken from actual market situations and not from an idealized experiment with random assignment of workers to firms. (3) In particular, if spillovers generate homogeneous firms, as they do with Equation 8, the estimate of [epsilon] from Equation 10 should be 1 regardless of the true [epsilon].

In the empirical work that follows, this problem can be solved through the use of data with a panel structure. Suppose one has data on [a.sub.jkt] and [A.sub.kt], for a cross section of workers and firms and a time series of periods. Then consider a regression of the form

(11) log([a.sub.jkt]) = [[pi].sub.j] + [epsilon] log([A.sub.kt]/[n.sub.kt]) + [[eta].sub.jkt],

which includes individual-level fixed effects [[pi].sub.j] as regressors. In this case, the estimate of [epsilon] is generated from the time series on each worker alone, comparing a single worker's productivity with changes in the composition of firms he has worked for. The variation used to measure the spillover is changes in the set of coworkers a given worker interacts with over his career. To the extent that there are some exogenous effects on firm composition, this approach uses only that variation to measure the spillover.

The second confounding factor related to regressions of Equation 10 is that they cannot distinguish productivity spillovers from exogenous firmwide effects on labor productivity. For instance, if a firm embarks on a capital deepening, every worker may see a similar change in productivity at the same time. This would lead to a positive omitted variables bias in the estimation of [epsilon]. In the empirical work that follows, the special structure of offensive production in basketball allows for several ways to check for such a bias.

Measuring Spillovers in Professional Basketball

Team sports provide a laboratory for the analysis of within-team spillovers. Individual-level productivity data are available and reliable, and players can be identified and matched to their teams. Many authors have extolled the usefulness of sports data in the empirical analysis of economic questions (e.g., Fort and Quirk 1995; Rosen and Sanderson 2001).

The National Basketball Association (NBA) is the largest and most popular professional basketball league in the United States. Player and team statistics are methodically recorded by official scorers present at each game. Each team plays 82 (nonplayoff) games during the season each year. Data for every player on every team during the 1988-1989 season through the 2000-2001 season, including a total of 5277 player-season observations, were taken from Jordan, Hubbard, and Stem (2001). Table 1 provides summary statistics for the NBA sample.

As discussed earlier, high-quality teammates provide value to players because they draw away the opposing team's defenders and provide a more unobstructed view of or pathway to the basket, allowing for higher productivity shot attempts. Thus, we expect [epsilon] to be positive for professional basketball. Note that I consider here only spillovers to offensive production, that is, scoring. While there may be spillovers in defense and other aspects of the game as well, offensive production in basketball is much easier to measure and allows for several robustness checks for omitted variables. (4)

The measure of player productivity corresponding to [a.sub.jkt] in the model is a "field goal scoring efficiency" statistic. It is calculated as a weighted average of a player's field goals and three-point shots and may be thought of as resembling a "points per shot attempt" statistic (divided by two) (5).

[Scoeff.sub.jkt] = F[G.sub.jkt] + 1/2 [Three.sub.jkt]/FG[A.sub.jkt],

where F[G.sub.jkt] is the number of successful (scoring) field goals (including three-point shots) scored by player j on team k in year t, FG[A.sub.jkt] is the total number of field goal attempts by player j on team k in year t, and [Three.sub.jkt] is the number of successful three-point shots scored by player j on team k in year t.

A version of Equation 11 can be estimated by least squares to derive an estimate of the spillover, [epsilon]:

(12) log [Scoeff.sub.jkt] = [[pi].sub.j] + [epsilon](log [Scoeff.sub.kt,-j]) + [beta][X.sub.jkt] + [v.sub.jkt],

where [[pi].sub.j] = ln [[phi].sub.j] is a fixed effect measuring player j's offensive talent and [X.sub.jkt] contains covariates. The term [Scoeff.sub.kt,-j] is the average (weighted by shot attempts) scoring efficiency for the rest of the team in season t, excluding player j. Player j's scoring efficiency is removed from that of his team on the right-hand side, relaxing an unrealistic (but mathematically convenient) implication of the previously mentioned theory, namely, that workers receive spillovers from their own production. If one were to include player j's scoring efficiency in the right-hand-side regressor, simultaneity bias would imply a spurious positive bias to the estimate of [epsilon] because each player's own efficiency is necessarily a component of the team's efficiency. (6)

Equation 12 was estimated with the observations weighted by the number of minutes a player was on the floor over the course of a season, (7) and the results are reported in the first column of Table 2.

Here, [X.sub.jkt] includes a quartic function in field goal attempts in order to hold constant the value of a player's marginal attempts as well as year fixed-effect dummy variables.

The number in the first row and first column of Table 2 may be interpreted to mean that, if a given player was traded to a team with 10% higher scoring efficiency, the player's own scoring efficiency would be expected to increase by 4.55%. (9)

Robustness Checks for Omitted Variables Bias

As discussed in the previous section, a major problem with empirical exercises like this one is the difficulty of separating true spillovers from firmwide effects that impact each worker individually. There are many examples of such effects that might confound the spillover effect estimated earlier. An illustrative case is that of coaching changes. Suppose for argument's sake that all players are equally talented; that is, [[phi].sub.j] is independent of j. However, suppose some teams have high-quality coaches while others have low-quality coaches. The high-quality coaches work individually with each player on their teams to improve player skills, while low-quality coaches do not work effectively with players. On a team that fires a low-quality coach and replaces him with a high-quality coach, all players would improve simultaneously, and a regression of the sort run earlier, omitting coach quality as a regressor, would produce a positive coefficient [epsilon] when the quality of a team's coach changed, even if no interaction among players existed.

Improvements in other teams' defensive skill, complementarities in merchandising of player jerseys, convexity in the objective function of the owner or players over games won, [10] "momentum" (see Scully 1995), good players' increased desire to win championships, physical changes in stadiums, changes in team experience levels, and changes in fan support levels could all potentially improve each individual player's productivity on the team simultaneously and so generate a similar omitted variables bias in the estimation of [epsilon].

Some of these objections could be individually tested with appropriate data, such as a dummy variable for coaching changes; however, this ad seriatum approach could never sufficiently answer more than a few objections. More ideal would be a simple test that would distinguish interaction spillovers from all other theories of complementarity and omitted variables. In the following, three such tests are proposed that answer many objections along the lines mentioned here.

First, as mentioned earlier, one can check for productivity spillovers using only data on flee throws. When a foul is committed, the player whom the foul was committed against is given the opportunity to take one or more shots while standing alone at a fixed distance from the basket. His teammates may verbally encourage him, and fans may try to distract him, but there is little interaction among players while these shots are taken. Thus, if teamwide shocks are causing positive bias in the measured [epsilon], a similar positive coefficient should also appear when regressing players' individual flee throw shooting accuracy against that of their teammates. (11) To check this possibility, a regression was run of the same form as Equation 12, but with flee throw percentages substituted for scoring efficiencies:

(13) log F[T%.sub.jkt], = [[pi].sub.j], + [[epsilon].sub.FT](log F[T%.sub.kt,-j]) + [beta][X.sub.jkt], + [v.sub.jkt].

F[T%.sub.jkt] = F[T.sub.jkt]/FT[A.sub.jkt], player j's free throw percentage, and F[T%.sub.kt,-j] is the average free throw efficiency (weighted by shot attempts) of the rest of the team, excluding player j. Now [X.sub.jkt] includes year fixed effects and a quartic equation in free throw attempts. (12)

The results of regressions in the form of Equation 13 are shown in the second column of Table 2. The measured "spillover" is near zero and statistically insignificant.

As a second test for omitted variables bias from teamwide shocks, Equation 12 was rerun, but instead of basketball data, data from professional baseball were used (Thorne, Palmer, and Gershman 2001). Offensive interaction among players in baseball is far less than that among basketball players. A baseball player bats alone. However, if the spillover in the basketball data were truly being driven by omitted variables instead of true player interaction, then we should expect to see in the baseball data a similarly high [epsilon].

Statistics on every player in the major leagues between the 1989 and 2000 seasons (inclusively) were used. As in professional basketball, an official scorer is present at every Major League Baseball game to compile individual and team data. There are several viable choices for a measure of individual offensive skill among baseball players. Here, however, I focus on one, on-base average, which is the percentage of times a player gets on base, via either hits or walks, out of all his at-bat opportunities.

The last column of Table 2 presents the results of running the same specification in Equation 12 on the baseball data set. Here the covariates include year fixed effects as well as a quartic function of at-bats (batting opportunities). As can be seen, the coefficient is much smaller than those derived from the basketball data and is statistically insignificant.

A third test that might distinguish between productivity spillovers and other factors in producing the results of the first row of Table 2 involves estimating again the basic Equation 12 but using different weighting schemes. If spillovers to productivity are actually generating the previous results, there should be more spillovers between players who play together more than between players who sit on the bench or rarely play. Previously, Equation 12 was estimated with a weighting approach using each player's minutes on the court over the season as weights. If Equation 12 were estimated without these weights and the spillover coefficient [epsilon] did not change, this would suggest that time on the court had no effect on the degree to which player quality spilled over and might introduce doubt about the degree to which spillovers were generating the data in the first row of Table 2. Table 3 presents the results of estimating Equation 12 under three different weighting schemes: with minutes played as weights, with no weighting, and with the number of games started during the season as weights. Fewer years of data are available in which the number of games started for each player are known. The coefficient with minutes played as weights is the same as the one estimated originally in Table 2. Both that coefficient and the coefficient weighting the data by the number of games the player started during the season are higher than that with no weighting. This too supports the hypothesis that productivity spillovers are driving the results derived in the first row of Table 2.

To summarize the previous results, it may be argued that the variation in a player's teammates over his career, used to measure spillovers among basketball players in the first column of Table 2, is not sufficiently exogenous for the result to be believable. However, to review the evidence presented here, if the spillover measured is the true parameter, the following predictions should hold:

(i) There should be no measurable spillover to free throw shooting,

(ii) there should be no measurable spillover to offensive production in baseball, and

(iii) weighting the data in favor of players who interact more should cause the measured spillover to increase.

All these conclusions are borne out in the data. If firmwide effects were driving the variation in teammates' productivity in the sample, such an effect would have to operate through some factor that is specific to basketball (and not baseball), correlated with the degree of player interaction, and that operates only through field goals and not through free throws. While some such factors undoubtedly exist, they seem relatively unlikely to explain the results presented here for basketball players.

Measuring Homogeneous Sorting in Professional Basketball

The previous section showed that there seem to be significant productivity spillovers between professional basketball players. Now we seek to know whether these spillovers generate complementarities, as in Equation 8, or not, as in Equation 9. If Equation 8 holds, one would expect a homogeneous sorting of players into teams. Thus, this section attempts to look for complementarities by answering the question, How much homogeneous sorting is there in the context of professional basketball?

To answer the question, a measure of the homogeneity of firms will also be needed. Consider the following statistic, H:

(14) H = [Var.sub.k] [log [A.sub.kt]]/Var[log [a.sub.jkt]]

The statistic in Equation 14 is the ratio of the cross-firm variance in productivity and the population variance in productivity. (13) In a case of perfect homogeneity, as implied by Equation 8, all workers in a given firm would be of equal productivity, so that all the variance in productivity would be between teams. In this case, H would take the value of 1. If there were perfect heterogeneity, all firms would have equal numbers of each type of worker, so the within-firm variance would be the same as the population variance, and the statistic in Equation 14 would take the value of 0.

However, H is biased by the existence of spillovers. If there are significant positive spillovers, even with random sorting, H will be positive since in this case the [a.sub.jk] on a team move together. Luckily, if the degree of spillover [epsilon] can be accurately measured from estimation of Equation 12, then it is possible to net out the size of the spillover from H, leaving the amount of homogeneity, exclusive of spillovers.

Let the degree of homogeneity caused by spillovers be [rho]. The Appendix derives the following relationship between H and [rho] (14):

(15) [rho] = H[(1 - [epsilon]).sup.2]/1 - [[epsilon].sup.2]H

The H-statistics discussed in Equation 14 were calculated separately for each of the years in the data set. The number presented in the first data column of Table 4 is the average of these. In most cases, H did not differ significantly across years. (15)

The second column in Table 4 replicates the estimated spillovers from Table 2. The final column of Table 4 performs the calculation suggested in Equation 15. Table 4 indicates that 22.06% of the variation in individual NBA basketball players' productivities can he explained by simply knowing the name of the team they play on. This suggests a significant amount of homogeneity. However, after accounting for spillovers, only 6.87 percentage points of this explanatory power could possibly be attributed to complementarities between players.

Rows 2 and 3 in Table 4 indicates that the degree to which complementarities between players can explain homogeneity in scoring efficiency is less than the degree of team homogeneity in free throw shooting and less than the degree of homogeneity in baseball teams. It seems that while spillovers to productivity may exist between basketball players, they do not generate complementarities as in Equation 8 but more closely resemble the form of Equation 9. In other words, the value of a good player, even including his spillover value, does not vary much across teams.

There are clearly other factors that also influence the composition of teams. For instance, it has been widely noted that cities with larger populations attract the best players because the aggregate value of wins for those teams to the population is larger, hence the relative dominance of New York--and Los Angeles--based teams in many competitive sports in North America. Moreover, [rho] may pick up some unobserved team characteristics, such as coaching quality and stadium facilities. The fact that there does seem to be some homogeneity in free throw shooting and in baseball among teams may encompass these factors. (16)

4. Conclusion

The model presented here gave conditions under which productivity spillovers across workers may or may not have equilibrium effects on the worker-firm matching process. In particular, if the spillover value of a worker is an increasing function of his coworkers' quality, the equilibrium sorting of workers into firms should generate firms homogeneous in worker quality.

Evidence from professional sports shows that, while there may be significant spillovers within basketball teams, these are not likely to have sorting effects.

In addition to providing evidence on spillovers and complementarities in professional sports, the empirical results presented here are also important in considering any attempt to measure players' marginal products, as discussed in section 1, or even in the "league-leaders" lists published in the sports sections of most daily newspapers. Further research might attempt to derive measures of players' "true" marginal products when productivity spillovers exist.

In addition, the results here suggest that spillovers are not a significant factor in determining team composition in the NBA and hence competitive balance in the league. Further research might determine and compare the importance of other factors, such as draft rules, size of fan base, and salary cap stringency, by regressing the H-statistic employed here on variables signifying these other factors.

Appendix

As noted in the text, the H-statistic described in Equation 14 includes measures of spillovers as well as the homogeneity caused by spillovers. The homogeneity component is reflected in the following statistic, given the assignment {[n.sub.jk]} in some period:

[rho] = [Var.sub.k][log [[summation of].sub.i][n.sub.jk][[phi].sub.jk]]/Var[log [[phi].sub.jk]].

[rho] cannot be calculated directly, since [[phi].sub.jk] is not observable. Only [a.sub.jk] is observable. However, [rho] can be derived indirectly if a measure of the spillover [epsilon] is available from estimation of Equation 15.

THEOREM: If Var([n.sub.k]) = 0 and players are of measure zero relative to their teams,

[rho] = H[(1 -[epsilon]).sup.2]/1 - [[epsilon].sup.2]H.

PROOF: Let Var([n.sub.kt]) = 0. Then the statistic in Equation 15 is:

(by Equation 8) [R.sup.2] = [Var.sub.k][log [A.sub.k]]/Var[log [a.sub.jk]] = [Var.sub.k][log [A.sub.k]]/Var[log [[phi].sub.j] + [epsilon] log [A.sub.k] - [epsilon] log [n.sub.k]]

= [Var.sub.k][log [A.sub.k]]/Var[log [[phi].sub.j]] + [[epsilon].sup.2]Var[log [A.sub.k]] + 2[epsilon]Cov[log [[phi].sub.j], log [A.sub.k]].

Thus,

[R.sup.2]{Var[log [[phi].sub.j]] + 2[epsilo]Cov[log [[phi].sub.j], log [A.sub.k]]} = (1 - [R.sup.2][[epsilon].sup.2])Var[log [A.sub.k]] = (1 - [R.sup.2][[epsilon].sup.2][(1/1 - [epsilon]).sup.2][Var.sub.k][log [summation over (j)] n[[phi].sub.jk]].

Now,

Cov[log[[phi].sub.jk], log [A.sub.k]] = Cov[log [[phi].sub.jk], -[epsilon]/1 - [epsilon] log [n.sub.k] + 1/1 - [epsilon] log [summation over (j)] [n.sub.jk][[phi].sub.jk]] = 1/1 - [epsilon]Cov[log [[phi].sub.jk], log [summation over (j)] [n.sub.jk] [[phi ].sub.jk]

So,

[R.sup.2]Var[log [[phi].sub.j]] - 1 [[epsilon].sup.2][R.sup.2]/[(1 - [epsilon]).sup.2]Var[log [summation over (j)] [n.sub.jk][[phi].sub.jk]] + 2[epsilon][R.sup.2]/1 - [epsilon]Cov[log [[phi].sub.j], log [summation over (j)] [n.sub.jk][[phi].sub.jk]] = 0,

which implies that

Var[log [[summation of].sub.j] [n.sub.jk][[phi].sub.jk]]/Var[log [[phi].sub.j]] = H[(1 - [epsilon]).sup.2]/1 - [[epsilon].sup.2]H + 2[epsilon]H/1 - H[[epsilon].sup.2] Cov[log[[phi].sub.jk], log [[summation of].sub.j] [n.sub.jk] [[phi].sub.jk]]/Var[log [[phi].sub.jk]

If players are of zero individual measure, the second term in the previous equation vanishes. QED.

The second term in the previous equation is the bias from Assumption 2. It is recognizable as the elasticity of total firm talent with respect to individual talent as estimated in a linear regression.

Table 1. NBA Basketball Summary Statistics

                               Mean (SD)

Scoring efficiency                 .5211
                                  (.0404)
Field goal % (excluding 3s)        .4802
                                  (.0427)
3-point %                          .2573
                                  (.1557)
3s made                          35.37
                                (45.57)
3s attempted                    100.47
                               (119.34)
Field goals made                337.98
                               (166.95)
Field goals attempted           724.69
                               (341.56)
Free throw %                       .7474
                                  (.0949)
Free throws made                175.23
                               (112.71)
Free throws attempted           231.99
                               (143.24)
Observations                   5277

Table 2. Spillovers in Sports

                                   Scoring Efficiency

Team average scoring efficiency         .4547 *
                                       (.0318)
Attempts                                .0005 *
                                      (4.80e-5)
Attempts (2)                          -6.75e-9 *
                                      (1.02e-7)
Attempts (3)                           4.20e-10 *
                                      (8.40e-12)
Attempts (4)                          -9.43e-14 *
                                      (2.36e-14)
Adjusted [R.sup.2]                      .5612
Observations                         5227

                                   Dependent Variables

                                       Free Throw%        One Base
                                                           Average

Team average scoring efficiency          .01195           .0752
                                        (.03565)         (.0398)
Attempts                                 .0010 *          .0024 *
                                        (.0001)          (.0002)
Attempts (2)                           -2.86e-6 *       -9.15e-6 *
                                       (5.57e-7)        (1.23e-6)
Attempts (3)                            4.00e-9 *        1.58e-8 *
                                       (1.01e-9)        (2.60e-9)
Attempts (4)                           -1.96e-12        -9.64e-12 *
                                       (5.92e-13        (1.87e-12)
Adjusted [R.sup.2]                       .7162            .6138
Observations                         5102             9185

* Signifies significance at the 5% level. The columns refer to
regressions of Equations 12, 13, and 14, from left to right,
respectively. In the first column, "Attempts" variables correspond to
field goal attempts, while in the second column, they correspond to
free throw attempts and in the third at-bats.

Table 3. Within-Team Spillovers to Scoring Efficiency in the NBA by
Weighting Schemes

                                     Weighting Scheme

                           None       Minutes Played      Starts

Team average scoring        .3919 *         .4547 *         .5003 *
  efficiency               (.0614)         (.0318)         (.0508)
Observations            5227            5227            1751
Adjusted [R.sup.2]          .5131           .5612           .7425

* Signifies significance at the 5% level. The columns refer to
regressions of Equation 12 using the different weighting schemes
indicated in each column.

Table 4. Homogeneity
                                           H      [epsilon]     [rho]

NBA scoring efficiency                   .2206      .4547       .0687
NBA free throw %                         .1109      .0120       .1083
Major League Baseball on-base average    .0841      .0752       .0720

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